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Primary Maths · Year 5Chapter 3 · Multiples, factors, primes, squares and cubes
Year 5 · Chapter 3

Year 5 Prime Numbers, Factors, Squares and Cubes: worked examples and questions with answers

By the end of this chapter you will be able to

  • Find multiples of a number, common multiples of two numbers and the lowest common multiple, and solve problems about things that repeat
  • Use the tests of divisibility for 2, 3, 4, 5, 6, 8, 9, 10, 25 and 100
  • Find all the factor pairs of a number in order, and find common factors and the highest common factor of two numbers
  • Know prime and composite numbers, remember the primes up to 19 and decide whether a number up to 100 is prime
  • Write a number up to 100 as a product of prime factors using a factor tree
  • Know square numbers (up to 12 × 12 and beyond) and cube numbers, and use the ² and ³ notation
  • Solve number-clue puzzles and reasoning problems: always, sometimes, never; prove it; spot the mistake

You should already know

  • Times tables up to 12 × 12 and the related division facts (Year 4)
  • Factor pairs and square numbers (Year 4 Chapter 3)
  • Counting in steps of 2, 3, 4, 5, 6, 7, 8, 9, 10, 25
  • Multiplying and dividing in your head (Chapters 4 and 5 build on this)

Where this leads

  • Multiplication with factors and squares (Chapter 4)
  • Equivalent fractions and simplifying (Chapter 6)
  • Area of squares and volume of cubes (Chapter 8)
  • Highest common factor, lowest common multiple and prime factorisation in Year 6 and secondary school

National Curriculum for England, Key Stage 2 mathematics, Year 5. Content: Identify multiples and factors, including finding all factor pairs of a number, and common factors of two numbers, Know and use the vocabulary of prime numbers, prime factors and composite (non-prime) numbers, Establish whether a number up to 100 is prime and recall prime numbers up to 19, Recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³).

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FWork out 7 × 8.
2
FCount in 6s. Write the missing numbers.
06+612+618+6?+6?+6?+6
3
FWork out 72 ÷ 9.
4
FWrite all the pairs of whole numbers that multiply to make 12.
5
FIs 25 a square number?   Write Yes or No.
3.1

Multiples and common multiples

Year 5
Identify multiples, Common multiples of two numbers, Lowest common multiple (introduced), Problems with multiples in context

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Multiples and common multiplesComing soon

A multiple of a number is an answer in that number’s times table. You find the multiples by counting on in steps of that number. Count in 4s: 4, 8, 12, 16, 20, 24… These are the multiples of 4. Each one is 4 × 1, 4 × 2, 4 × 3, and so on.

04+48+412+416+420+424+4
The first six multiples of 4. Each hop is +4.

Multiples never stop. There is no biggest multiple, so we always say “the first six multiples” or “the multiples up to 50”. A number is a multiple of itself (4 = 4 × 1). In this chapter we start at the number itself, not at 0. Some times tables have patterns: every multiple of 2 is even, every multiple of 5 ends in 0 or 5, and every multiple of 10 ends in 0.

Common multiples. Look at the hundred square. The multiples of 3 are blue. The multiples of 4 are orange. Some numbers are in both times tables. They are green. They are the common multiples of 3 and 4: 12, 24, 36, 48… The smallest of them is 12. It is called the lowest common multiple, or LCM.

12345678910
11121314151617181920
21222324252627282930
31323334353637383940
41424344454647484950
51525354555657585960
61626364656667686970
71727374757677787980
81828384858687888990
919293949596979899100
Blue: multiples of 3. Orange: multiples of 4. Green: multiples of both (common multiples). The LCM of 3 and 4 is 12.

Finding the LCM with lists. Write the multiples of each number. Keep going until a number is in both lists. That is the LCM. For 6 and 8: 6, 12, 18, 24 and 8, 16, 24. The LCM is 24. (Try not to just multiply the two numbers: 6 × 8 = 48 is a common multiple, but not the lowest one.)

Multiples in real life. Buses, lights and bells all repeat. If one bus leaves every 6 minutes and another every 8 minutes, they leave together again when the time is a common multiple of 6 and 8. The picture shows both buses. The first time they leave together is the LCM.

Bus A612182430364248Bus B81624324048minutes after they leave together
Bus A leaves every 6 minutes. Bus B leaves every 8 minutes. They leave together at 24 minutes and 48 minutes (green).

Key ideas

  • A multiple of a number is in its times table: you get it by multiplying the number by 1, 2, 3, 4…
  • To list multiples, count on in steps of the number. Every multiple is the same distance from the one before.
  • To test a number, look at the multiples just below and just above it. Or divide: if there is no remainder, it is a multiple.
  • A common multiple of two numbers is in both lists. The smallest one is the lowest common multiple (LCM).
  • For real-life problems with repeats (buses, lights), the answer is a common multiple.
Worked example 3.1.1
Write the first eight multiples of 7.

Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…

Method. The multiples of 7 are the answers in the 7 times table. Multiply 7 by 1, 2, 3, 4 and so on.

  1. Multiply 7 by 1, 2, 3, … Each answer is a multiple of 7.
    7 × 1 = 7    7 × 2 = 14    7 × 3 = 21
    7 × 4 = 28    7 × 5 = 35    7 × 6 = 42
    7 × 7 = 49    7 × 8 = 56
  2. Write the multiples in order. The first multiple of a number is the number itself.
    7, 14, 21, 28, 35, 42, 49, 56
  3. Look at the steps. The multiples go up by 7 each time, so each one is 7 more than the one before.
    7, 7 + 7 = 14, …, 56

Check. The last multiple is 7 × 8 = 56, and 56 ÷ 7 = 8. ✓

Watch out. Start at 7, not at 0. And do not add a different number each time: every step is +7.

Worked example 3.1.2
Is 56 a multiple of 8?

Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…

Method. A multiple of 8 is a number in the 8 times table. Find the multiples of 8 either side of the number and see if it is one of them.

  1. Look at the 8 times table near 56. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.
    8 × 7 = 56   and   8 × 8 = 64
  2. Compare with the number. Is it one of them?
    56 is equal to 56.
  3. Answer the question.
    56 is a multiple of 8: Yes

Check. Divide: 56 ÷ 8 = 7. There is no remainder. ✓

Watch out. A multiple is always in the times table. Being close (like 64 for 56) is not enough.

Worked example 3.1.3
Is 75 a multiple of 6?

Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…

Method. A multiple of 6 is a number in the 6 times table. Find the multiples of 6 either side of the number and see if it is one of them.

  1. Look at the 6 times table near 75. Multiples are spaced the same distance apart, so we only need the multiples just below and just above the number.
    6 × 12 = 72   and   6 × 13 = 78
  2. Compare with the number. Is it one of them?
    75 is between 72 and 78.
  3. Answer the question.
    75 is not a multiple of 6: No

Check. Divide: 75 ÷ 6 = 12 remainder 3. There is a remainder, so it is not exact. ✓

Watch out. A multiple is always in the times table. Being close (like 78 for 75) is not enough.

Worked example 3.1.4
Write all the multiples of 9 that are between 50 and 100.

Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…

Method. List the multiples of 9, and keep only the ones between 50 and 100.

  1. Count on in 9s until you pass 100. We need every multiple, so do not stop too soon.
    9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108
  2. Cross out the ones that are not between 50 and 100. “Between 50 and 100” does not include 50 or 100 themselves.
    45, 54, 63, 72, 81, 90, 99, 108
  3. Write the numbers that are left.
    54, 63, 72, 81, 90, 99

Check. The first one, 54, is more than 50, and 45 is not. The last, 99, is less than 100, and 108 is not. ✓

Watch out. Check the first and the last numbers against the limits.

Worked example 3.1.5
Write the common multiples of 3 and 4 that are 36 or less.

Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.

Method. Write the multiples of each number. The numbers that are in both lists are the common multiples.

  1. List the multiples of 3. Count on in 3s up to 36.
    3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36
  2. List the multiples of 4. Count on in 4s up to the same number.
    4, 8, 12, 16, 20, 24, 28, 32, 36
  3. Find the numbers in both lists. These are the common multiples. They are in bold.
    12, 24, 36
  4. Write the answer. Only list the common multiples up to 36.
    12, 24, 36

Check. Each one is in both times tables: 12 = 3 × 4 = 4 × 3, 24 = 3 × 8 = 4 × 6. ✓

Watch out. A number must be in both lists to be a common multiple.

Worked example 3.1.6
Find the lowest common multiple (LCM) of 6 and 8.

Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.

Method. Write the multiples of each number. The numbers that are in both lists are the common multiples.

  1. List the multiples of 6. Count on in 6s. Keep going until a number appears in both lists.
    6, 12, 18, 24
  2. List the multiples of 8. Count on in 8s up to the same number.
    8, 16, 24
  3. Find the numbers in both lists. These are the common multiples. They are in bold.
    24
  4. Pick the smallest. The lowest common multiple is the first number that is in both lists.
    The LCM of 6 and 8 is 24.

Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3. Both divide exactly. ✓

Watch out. The lowest common multiple is not always 6 × 8 = 48.

Worked example 3.1.7
The beach bus leaves the station every 12 minutes. The zoo bus leaves every 9 minutes. Both buses leave at 9:00 am. When do they next leave together?
Beach122436Zoo9182736minutes after 9:00 am

Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.

Method. Write the times for each one as multiples. The first time that is in both lists is the lowest common multiple.

  1. Write down when each one happens. They start together at 0. The beach bus leaves every 12 minutes and the zoo bus every 9 minutes.
    Every 12: 12, 24, 36
    Every 9: 9, 18, 27, 36
  2. Find the times that are in both lists. They happen together when the time is a common multiple of the two numbers.
    Together at: 36
  3. Pick the first one. The first time they are together again is the lowest common multiple.
    LCM of 12 and 9 = 36
  4. Add it to the start time. 36 minutes after 9:00 am.
    9:00 am + 36 minutes = 9:36 am

Check. 36 ÷ 12 = 3 and 36 ÷ 9 = 4, so both happen exactly at 36. ✓

Watch out. The answer is the LOWEST number in both lists. Do not stop at the first time one of them happens.

Worked example 3.1.8
A red light flashes every 4 seconds. A blue light flashes every 6 seconds. They flash together now. After how many seconds do they next flash together?

Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.

Method. Write the times for each one as multiples. The first time that is in both lists is the lowest common multiple.

  1. Write down when each one happens. They start together at 0. The red light flashs every 4 seconds and the blue light every 6 seconds.
    Every 4: 4, 8, 12
    Every 6: 6, 12
  2. Find the times that are in both lists. They happen together when the time is a common multiple of the two numbers.
    Together at: 12
  3. Pick the first one. The first time they are together again is the lowest common multiple.
    LCM of 4 and 6 = 12
  4. Write the answer.
    12 seconds after the start

Check. 12 ÷ 4 = 3 and 12 ÷ 6 = 2, so both happen exactly at 12. ✓

Watch out. The answer is the LOWEST number in both lists. Do not stop at the first time one of them happens.

Watch out

Do not mix up multiples and factors. Multiples are bigger (or equal): the multiples of 6 are 6, 12, 18… The factors of 6 are smaller (or equal): 1, 2, 3, 6. Also, the LCM is the lowest number in both lists. Multiplying the two numbers always gives a common multiple, but not always the lowest one.

Exercise 3A  Multiples and common multiples

Non-calculator
1
FWrite the first six multiples of each number.
(a)Multiples of 3
(b)Multiples of 4
(c)Multiples of 6
(d)Multiples of 8
(e)Multiples of 11
2
FCount on in 7s. Write the missing numbers on the number line.
07+7?+721+7?+735+7?+749+7
3
FIs each number a multiple of 6?   Write Yes or No.
(a)18
(b)40
(c)54
(d)72
(e)100
4
FWrite all the multiples that are between the two numbers.
(a)Multiples of 8 between 30 and 80
(b)Multiples of 12 between 50 and 100
(c)Multiples of 15 between 20 and 100
5
FThe multiples of 9 are shaded on the hundred square.
12345678910
11121314151617181920
21222324252627282930
31323334353637383940
41424344454647484950
51525354555657585960
61626364656667686970
71727374757677787980
81828384858687888990
919293949596979899100
(a)What is the next shaded number after 63?
(b)How many numbers on the grid are shaded?
(c)Is 108 a multiple of 9? (It is not on the grid.)
6
FFind the lowest common multiple (LCM) of each pair of numbers.
(a)3 and 5
(b)4 and 6
(c)6 and 9
(d)8 and 12
7
FThe Venn diagram shows the numbers from 1 to 36 that are multiples of 4 or multiples of 6.
Multiples of 4Multiples of 6481620283212243661830Not in either circle: 1, 2, 3, 5, 7, 9, 10, 11, …
(a)What do the numbers in the middle have in common?
(b)What is the lowest common multiple of 4 and 6?
(c)What is the next common multiple after 36?
8
PSThe Hull bus leaves a bus station every 15 minutes. The Leeds bus leaves every 20 minutes. Both leave at 8:00 am. At what time do they next leave together?
Hull15304560Leeds204060minutes after 8:00 am
9
PSA red light flashes every 4 seconds. A blue light flashes every 5 seconds. They flash together at the start. At what times in the next 60 seconds do they flash together?
Red4812162024283236404448525660Blue51015202530354045505560seconds
10
PSLeo counts in 8s: 8, 16, 24… He keeps going. Will he say each number?
(a)120
(b)150
(c)200
11
RZainab writes the first six multiples of 7:   7, 14, 21, 27, 35, 42.   One number is wrong. Find it and write the correct number.
12
RAlways, sometimes or never?
(a)A multiple of 4 is also a multiple of 2.
(b)A multiple of 3 is an odd number.
13
RChallenge!
(a)I am a multiple of 8 and a multiple of 6. I am between 80 and 100. What number am I?
(b)Mia says, “To find the lowest common multiple of two numbers, you just multiply them.” Use 6 and 9 to show that Mia is wrong.
3.2

Tests of divisibility

Year 5
Multiples and tests of divisibility: 2, 3, 4, 5, 6, 8, 9, 10, 25 and 100

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Tests of divisibilityComing soon

A number is divisible by another number if you can divide it exactly, with nothing left over. 24 is divisible by 6 because 24 ÷ 6 = 4. This is the same as saying that 24 is a multiple of 6. But what about a big number like 3,276? Is it divisible by 4? You do not need to divide. A test of divisibility gives a quick answer by looking at the digits.

A number is divisible by…if…Example
2The last digit is 0, 2, 4, 6 or 8 (an even number).3,846 ends in 6 → yes
3The digits add up to a number in the 3 times table.4,263: 4+2+6+3 = 15 → yes
4The last two digits make a number in the 4 times table. (Or halve twice.)2,716: 16 is in the 4s → yes
5The last digit is 0 or 5.1,235 ends in 5 → yes
6It is divisible by 2 and by 3.438 is even; 4+3+8 = 15 → yes
8Halve it three times and you always get a whole number. (Or the last three digits are in the 8s.)648 → 324 → 162 → 81 → yes
9The digits add up to a number in the 9 times table.7,893: 7+8+9+3 = 27 → yes
10The last digit is 0.5,430 ends in 0 → yes
25The last two digits are 00, 25, 50 or 75.1,375 ends in 75 → yes
100The last two digits are 00.4,500 ends in 00 → yes

Why they work. Every ten is even, and every ten is in the 5 times table, so only the last digit matters for 2, 5 and 10. Every hundred is in the 4 times table and the 25 times table, so only the last two digits matter for 4, 25 and 100. For 3 and 9 the sum of the digits tells you. The tests for 6 and 8 use the ones you already know: 6 = 2 × 3, and 8 means halving three times.

648324÷2162÷281÷2
Is 648 divisible by 8? Halve three times: 648, 324, 162, 81. We could halve it three times, so yes. (648 = 8 × 81.)
42+6+3+= 15
Is 4,263 divisible by 3? Add the digits: 4 + 2 + 6 + 3 = 15. 15 is in the 3 times table, so yes. (It is not in the 9 times table, so 4,263 is not divisible by 9.)

Key ideas

  • 2, 5, 10: look at the last digit.
  • 4, 25, 100: look at the last two digits (for 4: halve twice).
  • 8: halve three times and stay whole.
  • 3 and 9: add up all the digits. Is the total in the 3 (or 9) times table?
  • 6: divisible by 2 and by 3. (You can combine two tests when the numbers share no factor except 1: for 12 use 3 and 4.)
Worked example 3.2.1
Is 4,386 divisible by 2, by 5 or by 10?

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Use the test of divisibility for each number in turn. Write down which tests pass.

  1. Choose the tests. Each number has its own quick test.
    2, 5, 10
  2. Do each test on 4,386. Look only at the digits the test needs.
    TestWorkingDivisible?
    2The last digit of 4,386 is 6. Is it 0, 2, 4, 6 or 8? yes.yes
    5The last digit of 4,386 is 6. Is it 0 or 5? no.no
    10The last digit of 4,386 is 6. Is it 0? no.no
  3. List the numbers that pass.
    2

Check. Divide one to check: 4,386 ÷ 2 = 2,193. ✓

Watch out. Do every test. A number can pass some tests and fail others.

Worked example 3.2.2
Is 738 divisible by 3?

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Use the test of divisibility for 3. It saves doing the whole division.

  1. Remember the test for 3. Every 10, 100, 1000 is one more than a multiple of 3 (9, 99, 999), so only the digit total matters.
    Add up the digits. If the total is in the 3 times table, the number is divisible by 3.
  2. Use the test on 738. Only look at the digits that the test needs.
    7 + 3 + 8 = 18. 18 is in the 3 times table (3 × 6 = 18). yes.
  3. Answer the question.
    738 is divisible by 3: Yes

Check. Divide to check: 738 ÷ 3 = 246. No remainder. ✓

Watch out. Add all the digits, not just the last one.

Worked example 3.2.3
Is 516 divisible by 4?

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Use the test of divisibility for 4. It saves doing the whole division.

  1. Remember the test for 4. Every 100 is in the 4 times table, so only the last two digits matter.
    Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)
  2. Use the test on 516. Only look at the digits that the test needs.
    The last two digits are 16. 16 ÷ 4 = 4. yes.
  3. Answer the question.
    516 is divisible by 4: Yes

Check. Divide to check: 516 ÷ 4 = 129. No remainder. ✓

Watch out. Use the last two digits, not just the last one: 2,136 is divisible by 4 because 36 is.

Worked example 3.2.4
Is 7,240 divisible by 8?

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Use the test of divisibility for 8. It saves doing the whole division.

  1. Remember the test for 8. 8 is 2 × 2 × 2, so we can halve three times.
    Halve the number three times. If you get a whole number every time, it is divisible by 8.
  2. Use the test on 7,240. Only look at the digits that the test needs.
    7,240 ÷ 2 = 3,620 , 3,620 ÷ 2 = 1,810 , 1,810 ÷ 2 = 905. yes.
  3. Answer the question.
    7,240 is divisible by 8: Yes

Check. Divide to check: 7,240 ÷ 8 = 905. No remainder. ✓

Watch out. Halve three times. If you can only halve twice, the number is a multiple of 4 but not of 8.

Worked example 3.2.5
Is 534 divisible by 6?

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Use the test of divisibility for 6. It saves doing the whole division.

  1. Remember the test for 6. 6 is 2 × 3, so the number must pass both tests.
    The number is divisible by 2 and by 3.
  2. Use the test on 534. Only look at the digits that the test needs.
    Test 2: The last digit of 534 is 4. Is it 0, 2, 4, 6 or 8? yes.
    Test 3: 5 + 3 + 4 = 12. 12 is in the 3 times table (3 × 4 = 12). yes.
    Both tests must pass: yes.
  3. Answer the question.
    534 is divisible by 6: Yes

Check. Divide to check: 534 ÷ 6 = 89. No remainder. ✓

Watch out. Both tests must pass. A number that is only divisible by 2, or only by 3, is not divisible by 6.

Worked example 3.2.6
Is 6,372 divisible by 9?

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Use the test of divisibility for 9. It saves doing the whole division.

  1. Remember the test for 9. Every 10, 100, 1000 is one more than a multiple of 9 (9, 99, 999), so only the digit total matters.
    Add up the digits. If the total is in the 9 times table, the number is divisible by 9.
  2. Use the test on 6,372. Only look at the digits that the test needs.
    6 + 3 + 7 + 2 = 18. 18 is in the 9 times table (9 × 2 = 18). yes.
  3. Answer the question.
    6,372 is divisible by 9: Yes

Check. Divide to check: 6,372 ÷ 9 = 708. No remainder. ✓

Watch out. Add all the digits, not just the last one.

Worked example 3.2.7
Is 1,375 divisible by 25 or by 100?

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Use the test of divisibility for each number in turn. Write down which tests pass.

  1. Choose the tests. Each number has its own quick test.
    25, 100
  2. Do each test on 1,375. Look only at the digits the test needs.
    TestWorkingDivisible?
    25The last two digits are 75. Is it 00, 25, 50 or 75? yes.yes
    100The last two digits are 75. Is it 00? no.no
  3. List the numbers that pass.
    25

Check. Divide one to check: 1,375 ÷ 25 = 55. ✓

Watch out. Do every test. A number can pass some tests and fail others.

Worked example 3.2.8
Is 4,572 divisible by 12? Use the tests for 3 and 4.

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. 12 is 3 × 4, and 3 and 4 have no common factor except 1. So a number is divisible by 12 if it passes both the 3 test and the 4 test.

  1. Use the test for 3. Add up the digits. If the total is in the 3 times table, the number is divisible by 3.
    4 + 5 + 7 + 2 = 18. 18 is in the 3 times table (3 × 6 = 18). yes.
  2. Use the test for 4. Look at the last two digits. If they make a number in the 4 times table, the number is divisible by 4. (Or halve it twice.)
    The last two digits are 72. 72 ÷ 4 = 18. yes.
  3. Put the two tests together. It must pass both tests.
    yes and yes: 4,572 is divisible by 12.

Check. Divide: 4,572 ÷ 12 = 381. ✓

Watch out. This only works when the two numbers share no factor except 1. Do not use 2 and 4 to test for 8.

Worked example 3.2.9
Which of 2, 3, 4, 5, 6, 8, 9 and 10 is 360 divisible by?

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Use the test of divisibility for each number in turn. Write down which tests pass.

  1. Choose the tests. Each number has its own quick test.
    2, 3, 4, 5, 6, 8, 9, 10
  2. Do each test on 360. Look only at the digits the test needs.
    TestWorkingDivisible?
    2The last digit of 360 is 0. Is it 0, 2, 4, 6 or 8? yes.yes
    33 + 6 + 0 = 9. 9 is in the 3 times table (3 × 3 = 9). yes.yes
    4The last two digits are 60. 60 ÷ 4 = 15. yes.yes
    5The last digit of 360 is 0. Is it 0 or 5? yes.yes
    6Test 2: The last digit of 360 is 0. Is it 0, 2, 4, 6 or 8? yes.
    Test 3: 3 + 6 + 0 = 9. 9 is in the 3 times table (3 × 3 = 9). yes.
    Both tests must pass: yes.    		yes
    8360 ÷ 2 = 180 , 180 ÷ 2 = 90 , 90 ÷ 2 = 45. yes.yes
    93 + 6 + 0 = 9. 9 is in the 9 times table (9 × 1 = 9). yes.yes
    10The last digit of 360 is 0. Is it 0? yes.yes
  3. List the numbers that pass.
    2, 3, 4, 5, 6, 8, 9, 10

Check. Divide one to check: 360 ÷ 2 = 180. ✓

Watch out. Do every test. A number can pass some tests and fail others.

Watch out

For 4 you need the last two digits, not just the last one: 2,136 is divisible by 4 because 36 is. For 3 and 9 add all the digits. A number that ends in 5 is divisible by 5, but only some of them (the ones ending in 25 or 75, and 00, 50) are divisible by 25. And do not test for 8 by checking for 2 and 4: 12 passes both but is not divisible by 8.

Exercise 3B  Tests of divisibility

Non-calculator
1
FIs each number divisible by 2, by 5 or by 10?   Write which of these it is divisible by.
(a)1,230
(b)4,375
(c)508
(d)6,000
2
FAdd the digits to test for 3. Is each number divisible by 3?   Write Yes or No.
(a)51
(b)124
(c)372
(d)1,005
3
FAdd the digits to test for 9. Is each number divisible by 9?   Write Yes or No.
(a)63
(b)117
(c)4,321
(d)8,073
4
FIs each number divisible by 4?   Write Yes or No.
(a)124
(b)350
(c)1,516
(d)2,038
5
FHalve each number three times. Is it divisible by 8?   Write Yes or No.
(a)136
(b)420
(c)5,648
(d)7,300
6
FIs each number divisible by 25 or by 100?   Write which of these it is divisible by.
(a)450
(b)1,375
(c)3,600
(d)2,090
7
FIs each number divisible by 6?   Write Yes or No.
(a)42
(b)75
(c)138
(d)1,014
8
FWhich of these numbers are divisible by both 3 and 4?   24, 30, 36, 40, 48, 54, 60, 100
9
PSMia has a three-digit number: 4 ? 6.   The number is divisible by 3. Which digits could be in the box?
10
PSAisha has 756 stickers. Can she share them equally, with none left over, between…
(a)9 friends?
(b)4 friends?
(c)8 friends?
11
RZainab says, “2,136 is not divisible by 4, because 6 is not in the 4 times table.” Zainab is wrong. Explain her mistake and say whether 2,136 is divisible by 4.
12
RAlways, sometimes or never?
(a)A number divisible by 9 is divisible by 3.
(b)A number divisible by 3 is divisible by 9.
13
RLeo says, “A number is divisible by 6 if its last digit is 6.” Is Leo right? Give an example to show.
14
RZara has three digit cards: 3, 4 and 5. She says, “I can use all three cards to make a number that is divisible by 4.” Is Zara right? Show how you know.
3.3

Factors, factor pairs and common factors

Year 5
Identify factors, All the factor pairs of a number, Common factors of two numbers, Highest common factor (introduced)

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Factors, factor pairs and common factorsComing soon

A factor of a number is a whole number that divides into it exactly. 3 is a factor of 12 because 12 ÷ 3 = 4 with nothing left over. Two factors that multiply to make the number are a factor pair: 3 × 4 = 12, so 3 and 4 are a pair. You can see factor pairs in arrays. Twelve squares can make a 1 by 12, a 2 by 6 or a 3 by 4 rectangle, so the factor pairs of 12 are 1 × 12, 2 × 6 and 3 × 4. The factors of 12 are 1, 2, 3, 4, 6 and 12.

1 × 122 × 63 × 4
Three arrays show the three factor pairs of 12. (Turning an array round, like 6 × 2, gives the same pair.)

Finding all the factors, step by step. Do not just guess. Use this method.

  1. Start with 1 × the number. That is always a pair.
  2. Try 2. Does it divide exactly? If yes, write the pair. Then try 3, then 4, and so on, in order.
  3. Stop when the pairs meet: when the next number you would try is already in a pair, or its partner is a number you have already tried.
  4. Write all the numbers from the pairs in order. These are the factors.

A factor rainbow shows the pairs. Each arc joins two factors that multiply to make the number. 24 = 1 × 24, 2 × 12, 3 × 8 and 4 × 6.

1234681224
The factor rainbow of 24. The outside arc joins 1 and 24. The inside arc joins 4 and 6.

If the number is a square number, like 36, the middle pair is the same number twice (6 × 6). Write it once. That is why square numbers have an odd number of factors.

Common factors. Now look at two numbers. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. The numbers that are in both lists are the common factors: 1, 2, 3 and 6. The biggest one is the highest common factor (HCF). The HCF of 12 and 18 is 6. A Venn diagram is a good way to show this.

Factors of 12Factors of 184121236918
The middle holds the common factors of 12 and 18. The biggest is 6, so the HCF is 6.

Key ideas

  • A factor divides into a number exactly. A factor pair multiplies to make the number.
  • List factors in order: 1 × the number, then try 2, 3, 4… Stop when the pairs meet. Always include 1 and the number itself.
  • Factors come in pairs, so most numbers have an even number of factors. A square number has an odd number (the middle pair is one number).
  • A common factor of two numbers is a factor of both. The biggest is the highest common factor (HCF).
  • 1 is a factor of every number, so it is always a common factor.
Worked example 3.3.1
Is 7 a factor of 56?

Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

Method. A factor divides into the number exactly. Divide 56 by 7 and look for a remainder.

  1. Divide 56 by 7. Use your times tables.
    7 × 8 = 56 and 7 × 9 = 63
  2. Look at the remainder. No remainder means it divides exactly.
    56 ÷ 7 = 8 exactly
  3. Answer the question.
    7 is a factor of 56: Yes

Check. 7 × 8 = 56, so 7 and 8 are a factor pair. ✓

Watch out. A factor is smaller than (or equal to) the number. A multiple is bigger than (or equal to) it. Do not mix them up.

Worked example 3.3.2
Write all the factor pairs of 12.
1 × 122 × 63 × 4

Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

Method. List the factor pairs in order, trying 1, 2, 3, 4, … one at a time. Stop when the pairs meet.

  1. Start with 1 and the number itself. Every number has the factor pair 1 × itself.
    1 × 12 = 12  ✓  pair: 1 and 12
  2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
    2 × 6 = 12  ✓  pair: 2 and 6
    3 × 4 = 12  ✓  pair: 3 and 4
  3. Stop when the pairs meet. You do not need to try bigger numbers.
    The next number to try is 4, but 4 × 4 = 16 is more than 12. Its partner would be smaller than 4, and we have already tried those. The pairs have met.
  4. Write the factor pairs. Each pair multiplies to make the number.
    1 × 12, 2 × 6, 3 × 4

Check. Every pair multiplies to make 12. The factors come in pairs, so there are 6 of them. ✓

Watch out. Always include 1 and the number itself. Do not stop too early: try every whole number in order.

Worked example 3.3.3
Write all the factors of 30.

Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

Method. List the factor pairs in order, trying 1, 2, 3, 4, … one at a time. Stop when the pairs meet.

  1. Start with 1 and the number itself. Every number has the factor pair 1 × itself.
    1 × 30 = 30  ✓  pair: 1 and 30
  2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
    2 × 15 = 30  ✓  pair: 2 and 15
    3 × 10 = 30  ✓  pair: 3 and 10
    30 ÷ 4 = 7 remainder 2  ✗  these do not divide exactly
    5 × 6 = 30  ✓  pair: 5 and 6
  3. Stop when the pairs meet. You do not need to try bigger numbers.
    The next number to try is 6, but 6 × 6 = 36 is more than 30. Its partner would be smaller than 6, and we have already tried those. The pairs have met.
  4. Write the factors in order. Take the smaller numbers from each pair and the bigger ones, and put them in order.
    1, 2, 3, 5, 6, 10, 15, 30

Check. Every pair multiplies to make 30. The factors come in pairs, so there are 8 of them. ✓

Watch out. Always include 1 and the number itself. Do not stop too early: try every whole number in order.

Worked example 3.3.4
Write all the factors of 36.
123469121836

Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

Method. List the factor pairs in order, trying 1, 2, 3, 4, … one at a time. Stop when the pairs meet.

  1. Start with 1 and the number itself. Every number has the factor pair 1 × itself.
    1 × 36 = 36  ✓  pair: 1 and 36
  2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
    2 × 18 = 36  ✓  pair: 2 and 18
    3 × 12 = 36  ✓  pair: 3 and 12
    4 × 9 = 36  ✓  pair: 4 and 9
    36 ÷ 5 = 7 remainder 1  ✗  these do not divide exactly
    6 × 6 = 36  ✓  the same number twice, so write 6 only once
  3. Stop when the pairs meet. You do not need to try bigger numbers.
    The pairs have met at 6 × 6. Any bigger number would pair with one we have already found.
  4. Write the factors in order. Take the smaller numbers from each pair and the bigger ones, and put them in order.
    1, 2, 3, 4, 6, 9, 12, 18, 36

Check. Every pair multiplies to make 36. The factors come in pairs, so there are 9 of them (an odd number, because the middle pair is the same number twice). ✓

Watch out. Always include 1 and the number itself. Do not stop too early: try every whole number in order.

Worked example 3.3.5
Write all the factors of 48.

Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

Method. List the factor pairs in order, trying 1, 2, 3, 4, … one at a time. Stop when the pairs meet.

  1. Start with 1 and the number itself. Every number has the factor pair 1 × itself.
    1 × 48 = 48  ✓  pair: 1 and 48
  2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
    2 × 24 = 48  ✓  pair: 2 and 24
    3 × 16 = 48  ✓  pair: 3 and 16
    4 × 12 = 48  ✓  pair: 4 and 12
    48 ÷ 5 = 9 remainder 3  ✗  these do not divide exactly
    6 × 8 = 48  ✓  pair: 6 and 8
  3. Stop when the pairs meet. You do not need to try bigger numbers.
    The next number to try is 7, but 7 × 7 = 49 is more than 48. Its partner would be smaller than 7, and we have already tried those. The pairs have met.
  4. Write the factors in order. Take the smaller numbers from each pair and the bigger ones, and put them in order.
    1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Check. Every pair multiplies to make 48. The factors come in pairs, so there are 10 of them. ✓

Watch out. Always include 1 and the number itself. Do not stop too early: try every whole number in order.

Worked example 3.3.6
Write the common factors of 12 and 18.
Factors of 12Factors of 184121236918

Key words. A common factor of two numbers is a number that is a factor of both. The highest common factor (HCF) is the biggest common factor.

Method. Write the factors of each number. The numbers in both lists are the common factors. The biggest one is the highest common factor (HCF).

  1. List the factors of 12. Use factor pairs.
    1 × 12, 2 × 6, 3 × 4
    Factors of 12: 1, 2, 3, 4, 6, 12
  2. List the factors of 18. Use factor pairs.
    1 × 18, 2 × 9, 3 × 6
    Factors of 18: 1, 2, 3, 6, 9, 18
  3. Find the numbers in both lists. These are the common factors (in bold).
    1, 2, 3, 6
  4. Write the answer. Always include 1: it is a factor of every number.
    1, 2, 3, 6

Check. 12 ÷ 6 = 2 and 18 ÷ 6 = 3, both exact. ✓

Watch out. 1 is always a common factor. The HCF is the biggest number in both lists, not the sum or the product.

Worked example 3.3.7
Find the highest common factor (HCF) of 24 and 36.

Key words. A common factor of two numbers is a number that is a factor of both. The highest common factor (HCF) is the biggest common factor.

Method. Write the factors of each number. The numbers in both lists are the common factors. The biggest one is the highest common factor (HCF).

  1. List the factors of 24. Use factor pairs.
    1 × 24, 2 × 12, 3 × 8, 4 × 6
    Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  2. List the factors of 36. Use factor pairs.
    1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
    Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  3. Find the numbers in both lists. These are the common factors (in bold).
    1, 2, 3, 4, 6, 12
  4. Pick the biggest. The highest common factor is the biggest common factor.
    HCF of 24 and 36 = 12

Check. 24 ÷ 12 = 2 and 36 ÷ 12 = 3, both exact. ✓

Watch out. 1 is always a common factor. The HCF is the biggest number in both lists, not the sum or the product.

Worked example 3.3.8
Ali has 24 red counters and 36 blue counters. He puts them into identical bags. Every counter is used. What is the greatest number of bags he can make?

Key words. A common factor of two numbers is a number that is a factor of both. The highest common factor (HCF) is the biggest common factor.

Method. Use common factors. The number of equal groups must divide into both amounts exactly, and we want the biggest possible number of groups.

  1. Say what the number of bags must be. The counters are put in identical bags with none left over. So the number of bags divides exactly into 24 and into 36.
    A factor of 24 and a factor of 36: a common factor.
  2. List the factors of 24 and of 36. Use factor pairs.
    Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
    Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  3. Find the biggest common factor. We want the greatest number of bags, so we want the highest common factor.
    Common factors: 1, 2, 3, 4, 6, 12
    HCF = 12
  4. Write the answer.
    The most bags is 12.

Check. 12 × 2 = 24 and 12 × 3 = 36. ✓

Watch out. The answer must be a factor of both numbers. The HCF is the biggest one.

Watch out

Do not stop after the first few pairs, and do not forget the biggest pair 1 × the number. For 36 the middle pair 6 × 6 gives one factor (6), not two. Also, a factor is never bigger than the number: 24 is a multiple of 12, not a factor of 12.

Exercise 3C  Factors and factor pairs

Non-calculator
1
FIs each number a factor of 48?   Write Yes or No.
(a)5
(b)6
(c)8
(d)9
(e)12
2
FThe arrays show 18 squares arranged in rectangles.
1 × 182 × 93 × 6
(a)Write the factor pair shown by each array.
(b)Write all the factors of 18.
3
FWrite all the factor pairs of each number.
(a)10
(b)14
(c)20
(d)28
4
FWrite all the factors of each number, in order.
(a)15
(b)21
(c)32
(d)40
5
FThis factor rainbow shows all the factors of 30. Write the two missing factors.
1?35?101530
6
FThese are square numbers. Write all their factors. Write the middle factor once.
(a)25
(b)49
(c)64
(d)100
7
FWrite all the factors of each number.
(a)56
(b)72
8
PSThere are 28 children in a PE lesson. The teacher splits them into equal teams. Nobody is left out. Each team must have more than 1 child, and there must be more than 1 team. How many children could be in each team?
9
PSMei has 36 square tiles. She uses all of them to make a rectangle. How many different rectangles can she make? (A rectangle turned on its side is the same rectangle.)
10
PSWhich number has more factors, 40 or 45? Write how many factors each number has.
11
RLiam lists the factors of 24:   1, 2, 3, 4, 6, 12, 24.   He has missed one factor. Which one?
12
RTrue or false?
(a)1 is a factor of every whole number.
(b)A factor of 20 can be bigger than 20.
(c)All the factors of an even number are even.
13
RAlways, sometimes or never?   A square number has an odd number of factors.   Test 16, 25 and 36 and explain your answer.

Exercise 3D  Common factors and the highest common factor

Non-calculator
1
FThe Venn diagram shows the factors of 18 and the factors of 24.
Factors of 18Factors of 249181236481224
(a)Write the common factors of 18 and 24.
(b)What is the highest common factor of 18 and 24?
(c)Why is 9 not in the middle?
2
FWrite the common factors of each pair of numbers.
(a)12 and 20
(b)15 and 25
(c)16 and 24
(d)18 and 30
3
FFind the highest common factor (HCF) of each pair of numbers.
(a)9 and 12
(b)14 and 21
(c)20 and 50
(d)36 and 48
4
FWhich of these numbers are common factors of 30 and 45?   3, 5, 6, 9, 15
5
PSA rectangular floor is 60 cm long and 45 cm wide. Square tiles are used to cover it with no gaps and no cutting.
(a)What is the side length of the biggest square tile that can be used?
(b)How many of these tiles are needed?
6
PSNoah has 18 apples and 30 oranges. He puts them in identical bags with no fruit left over.
(a)What is the greatest number of bags he can fill?
(b)How much of each fruit is in one bag?
7
RMia says, “The highest common factor of 12 and 18 is 36, because 12 × 3 = 36 and 18 × 2 = 36.” Mia has made a mistake. Explain it and find the HCF.
8
RWhich pair of numbers has only 1 as a common factor?   A  8 and 15    B  6 and 9    C  14 and 21    D  10 and 25
9
RChallenge!   Two different numbers less than 20 have a highest common factor of 6 and a lowest common multiple of 36. What are the two numbers?
3.4

Prime and composite numbers

Year 5
Prime numbers, Composite numbers, Recall primes to 19, Establish whether a number up to 100 is prime

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Prime and composite numbersComing soon

Look at the factors of 7: only 1 and 7. Look at the factors of 8: 1, 2, 4 and 8. A number like 7 is special. It is called a prime number. A prime number has exactly two factors: 1 and itself. A number like 8 has more than two factors, so it is a composite number. With square tiles you can see the difference. A prime can only make one rectangle: a single long row. A composite number can make more than one.

1 × 7
7 squares make only one rectangle (1 × 7). So 7 is prime.
1 × 82 × 4
8 squares make two different rectangles (1 × 8 and 2 × 4). So 8 is composite.

Two special numbers. The number 1 has only one factor, so it is neither prime nor composite. The number 2 is prime (factors 1 and 2), and it is the only even prime, because every other even number has 2 as a factor as well as 1 and itself.

Learn the primes up to 19: 2, 3, 5, 7, 11, 13, 17 and 19.

The Sieve of Eratosthenes. This is an old way to find primes, using a hundred square. Cross out 1. Keep 2, then cross out every multiple of 2. Keep 3, then cross out every multiple of 3. Keep 5 and cross out its multiples. Keep 7 and cross out its multiples. All the numbers that are left are the primes.

12345678910
11121314151617181920
21222324252627282930
31323334353637383940
41424344454647484950
51525354555657585960
61626364656667686970
71727374757677787980
81828384858687888990
919293949596979899100
Part-way through the sieve. 1 is purple. The multiples of 2 (orange) and of 3 (green) are shaded. 2 and 3 are kept (gold).
12345678910
11121314151617181920
21222324252627282930
31323334353637383940
41424344454647484950
51525354555657585960
61626364656667686970
71727374757677787980
81828384858687888990
919293949596979899100
The finished sieve. The 25 gold numbers are the primes up to 100: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Is a number up to 100 prime? You do not need a sieve every time. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them divides the number, it is not prime. If none of them does, the number is prime. Why only these four? Because 11 × 11 = 121 is bigger than 100, so any number up to 100 that is not prime must have a factor of 2, 3, 5 or 7.

Key ideas

  • A prime number has exactly two factors: 1 and itself. A composite number has more than two factors.
  • 1 is neither prime nor composite. 2 is the only even prime.
  • Primes up to 19: 2, 3, 5, 7, 11, 13, 17, 19. There are 25 primes up to 100.
  • To test a number up to 100: use the tests for 2, 3, 5 and 7. If none works, it is prime.
  • The Sieve of Eratosthenes crosses out the multiples of 2, 3, 5 and 7. The numbers left are prime.
Worked example 3.4.1
Is 13 prime, composite or neither? List its factors.

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. Find all the factors, count them, and decide.

  1. List the factors of 13. Use factor pairs.
    1 × 13
    Factors: 1, 13
  2. Count the factors.
    2 factors
  3. Decide. Prime: exactly two factors. Composite: more than two. 1: only one.
    13 is prime. Exactly two factors, so prime.

Check. Every number except 1 is either prime or composite. 13 has 2 factors. ✓

Watch out. 1 is not prime. 2 is prime (it has exactly two factors).

Worked example 3.4.2
Is 21 prime, composite or neither? List its factors.

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. Find all the factors, count them, and decide.

  1. List the factors of 21. Use factor pairs.
    1 × 21, 3 × 7
    Factors: 1, 3, 7, 21
  2. Count the factors.
    4 factors
  3. Decide. Prime: exactly two factors. Composite: more than two. 1: only one.
    21 is composite. More than two factors, so composite.

Check. Every number except 1 is either prime or composite. 21 has 4 factors. ✓

Watch out. 1 is not prime. 2 is prime (it has exactly two factors).

Worked example 3.4.3
Is 1 a prime number?

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. A prime number has exactly two factors. Find the factors of 1.

  1. List the factors of 1. Only 1 divides into 1.
    Factors of 1: 1
  2. Count the factors. A prime needs exactly two different factors.
    1 has only one factor.
  3. Decide. A composite number has more than two factors, so 1 is not composite either.
    1 is neither prime nor composite.

Check. The first prime is 2, with factors 1 and 2. ✓

Watch out. 1 is not a prime number, even though it looks like one.

Worked example 3.4.4
Is 51 a prime number?

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.

  1. Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.
    We look for a factor that is not 1 or 51
  2. Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.
    Is it divisible by 2? The last digit of 51 is 1. Is it 0, 2, 4, 6 or 8? no.
    Is it divisible by 3? 5 + 1 = 6. 6 is in the 3 times table (3 × 2 = 6). yes.
  3. Decide.
    51 = 3 × 17, so 51 has other factors. 51 is not prime.

Check. 3 × 17 = 51. ✓

Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.

Worked example 3.4.5
Is 91 a prime number?

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.

  1. Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.
    We look for a factor that is not 1 or 91
  2. Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.
    Is it divisible by 2? The last digit of 91 is 1. Is it 0, 2, 4, 6 or 8? no.
    Is it divisible by 3? 9 + 1 = 10. 10 is not in the 3 times table (9 and 12 are either side). no.
    Is it divisible by 5? The last digit of 91 is 1. Is it 0 or 5? no.
    Is it divisible by 7? 7 × 13 = 91 and 7 × 14 = 98. yes.
  3. Decide.
    91 = 7 × 13, so 91 has other factors. 91 is not prime.

Check. 7 × 13 = 91. ✓

Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.

Worked example 3.4.6
Is 97 a prime number?

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.

  1. Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.
    We look for a factor that is not 1 or 97
  2. Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.
    Is it divisible by 2? The last digit of 97 is 7. Is it 0, 2, 4, 6 or 8? no.
    Is it divisible by 3? 9 + 7 = 16. 16 is not in the 3 times table (15 and 18 are either side). no.
    Is it divisible by 5? The last digit of 97 is 7. Is it 0 or 5? no.
    Is it divisible by 7? 7 × 13 = 91 and 7 × 14 = 98. no.
  3. Decide.
    None of 2, 3, 5 or 7 divides 97, so 97 is prime.

Check. The only factor pair of 97 is 1 × 97. ✓

Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.

Worked example 3.4.7
Write all the prime numbers between 20 and 40.

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. Cross out the even numbers first. Then test each odd number that is left with the 3, 5 and 7 times tables.

  1. Write the numbers from 20 to 40.
    20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40
  2. Cross out the even numbers. Even numbers (except 2) are divisible by 2, so they are not prime.
    20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40
  3. Test each odd number that is left. Try the 3, 5 and 7 times tables. Stop when one divides it.
    21 = 3 × 7: not prime
    23: not in the 3, 5 or 7 times tables: prime
    25 = 5 × 5: not prime
    27 = 3 × 9: not prime
    29: not in the 3, 5 or 7 times tables: prime
    31: not in the 3, 5 or 7 times tables: prime
    33 = 3 × 11: not prime
    35 = 5 × 7: not prime
    37: not in the 3, 5 or 7 times tables: prime
    39 = 3 × 13: not prime
  4. Write the prime numbers.
    23, 29, 31, 37

Check. Each prime has no factor except 1 and itself. 4 primes between 20 and 40. ✓

Watch out. Do not forget to test with 7. For example 91 = 7 × 13.

Worked example 3.4.8
Use the sieve in this section to write the prime numbers between 60 and 80.

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. The prime numbers are the gold squares on the finished sieve. Look along the rows for 61 to 79.

  1. Find the rows for the 60s and 70s. The 60s are the seventh row and the 70s are the eighth row.
    61, 62, 63, …, 69, 70  and  71, 72, 73, …, 79, 80
  2. Read the gold numbers. Gold numbers have not been crossed out, so they are prime.
    60s: 61, 67    70s: 71, 73, 79
  3. Write the answer. Check that none of them is 60 or 80: they are not in the list.
    61, 67, 71, 73, 79

Check. Test one with the divisibility tests: 79 is odd, 7 + 9 = 16 is not in the 3s, it does not end in 0 or 5, and 7 × 11 = 77 and 7 × 12 = 84, so it is not in the 7s. 79 is prime. ✓

Watch out. Between 60 and 80 does not include 60 or 80. The primes are never even (except 2) so you can skip every even number.

Watch out

Do not forget that 1 is not prime, and that 2 is prime (even though it is even). Also, numbers that are odd are not always prime: 9, 15, 21, 27 and 33 are odd but composite. And test with 7: 49, 77 and 91 are not in the 2, 3 or 5 times tables, but they are multiples of 7.

Exercise 3E  Prime and composite numbers

Non-calculator
1
FWrite the factors of each number. Then write whether the number is prime, composite or neither.
(a)1
(b)7
(c)9
(d)11
(e)15
(f)17
2
FWrite down the prime numbers in this list.   2, 9, 11, 15, 17, 21, 23, 27
3
FRemember the prime numbers up to 19.
(a)Write all the prime numbers up to 19.
(b)Fill in the missing primes:   2, 3, ?, 7, ?, 13, ?, 19
4
FMaya is making a sieve. She has shaded the multiples of 2, 3 and 5 (but not the numbers 2, 3 and 5). The numbers that are not shaded are still to be tested.
12345678910
11121314151617181920
21222324252627282930
31323334353637383940
41424344454647484950
51525354555657585960
61626364656667686970
71727374757677787980
81828384858687888990
919293949596979899100
(a)Some numbers that are not shaded are not prime. Write them. (Do not include 1.)
(b)What do these numbers have in common?
5
FUse the tests of divisibility for 2, 3, 5 and 7. Is each number prime?   Write Prime or Not prime.
(a)39
(b)53
(c)57
(d)71
(e)87
(f)89
6
FWrite all the prime numbers between 30 and 50.
7
FEach of these numbers is made by multiplying two prime numbers. Find the two primes.
(a)35
(b)77
(c)91
8
PSI am a prime number between 30 and 50. My digits add up to 7. What number am I?
9
RMia says, “Every odd number is a prime number.” Is Mia right? Give an example to show.
10
RAlways, sometimes or never?
(a)A prime number is odd.
(b)A number with exactly two factors is a prime number.
(c)The sum of two prime numbers is a prime number.
11
RWhich number is the odd one out? Explain.   11    13    17    21
12
RProve it!   Explain why no even number bigger than 2 is a prime number.
13
RAmira says, “51 is prime, because it is not in the 2, 5 or 10 times tables.” Is Amira right?
14
RChallenge!   Find three pairs of prime numbers that add up to 30.
3.5

Prime factors and factor trees

Year 5
Prime factors, Writing a number as a product of prime factors, Factor trees (numbers up to 100)

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Prime factors and factor treesComing soon

The factors of 12 are 1, 2, 3, 4, 6 and 12. Two of them are prime: 2 and 3. They are the prime factors of 12. Every number bigger than 1 can be built by multiplying prime numbers. For example 12 = 2 × 2 × 3. This is called writing the number as a product of prime factors.

A factor tree helps you find them. Write the number at the top. Split it into any factor pair. Each branch that ends in a prime is finished (we circle it in gold). Any branch that is not prime is split again. Stop when every branch ends in a prime.

322412
A factor tree for 12. 12 = 3 × 4, then 4 = 2 × 2. The primes at the bottom are 3, 2 and 2, so 12 = 2 × 2 × 3.

Do different trees give different answers? No! You can start with any factor pair. These two trees start differently, but the primes at the bottom are the same: 2, 2, 3 and 3. Every number has only one set of prime factors.

22433936
23623636
Two trees for 36. Left: 36 = 4 × 9. Right: 36 = 6 × 6. Both end in 2, 2, 3, 3.

When you write the answer, put the primes in order, smallest first, with × between them: 36 = 2 × 2 × 3 × 3. For numbers up to 100 the only primes you will need are 2, 3, 5 and 7. You can check the answer by multiplying the primes back together.

A short way. Instead of 2 × 2 × 3 × 3 you can write 2² × 3². The little number says how many times the prime is used. You will meet this a lot in secondary school.

Key ideas

  • A prime factor is a factor that is a prime number.
  • A factor tree splits a number into factor pairs. Stop on primes. Circle every prime at the bottom.
  • You can start with any factor pair. The primes at the bottom are always the same.
  • Write the answer with the primes in order: 60 = 2 × 2 × 3 × 5.
  • Check by multiplying the primes together. You should get the number you started with.
Worked example 3.5.1
Write 12 as a product of prime factors. Use a factor tree.

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Split the number into factors with a factor tree. Keep going until every branch ends in a prime number. The primes at the ends are the prime factors.

  1. Split the number into a factor pair. Pick any factor pair you know from the times tables.
    12 = 3 × 4
    322412
  2. Keep splitting every number that is not prime. A number that is not prime can be split again. Stop on a prime: it cannot be split.
    4 = 2 × 2
  3. Circle the primes at the ends of the branches. These are the prime factors.
    3, 2, 2
  4. Write them in order, joined with ×. The smallest prime goes first.
    12 = 2 × 2 × 3

Check. Multiply the primes back together: 2 × 2 = 4, 4 × 3 = 12. ✓

Watch out. Every branch must end in a prime. 9 is not prime (3 × 3), and 1 is never used. You can start with any factor pair: the primes at the end are always the same.

Worked example 3.5.2
Write 30 as a product of prime factors.

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Split the number into factors with a factor tree. Keep going until every branch ends in a prime number. The primes at the ends are the prime factors.

  1. Split the number into a factor pair. Pick any factor pair you know from the times tables.
    30 = 2 × 15
    2351530
  2. Keep splitting every number that is not prime. A number that is not prime can be split again. Stop on a prime: it cannot be split.
    15 = 3 × 5
  3. Circle the primes at the ends of the branches. These are the prime factors.
    2, 3, 5
  4. Write them in order, joined with ×. The smallest prime goes first.
    30 = 2 × 3 × 5

Check. Multiply the primes back together: 2 × 3 = 6, 6 × 5 = 30. ✓

Watch out. Every branch must end in a prime. 9 is not prime (3 × 3), and 1 is never used. You can start with any factor pair: the primes at the end are always the same.

Worked example 3.5.3
Write 36 as a product of prime factors. Start the tree with 36 = 4 × 9.

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Split the number into factors with a factor tree. Keep going until every branch ends in a prime number. The primes at the ends are the prime factors.

  1. Split the number into a factor pair. Pick any factor pair you know from the times tables.
    36 = 4 × 9
  2. Keep splitting every number that is not prime. A number that is not prime can be split again. Stop on a prime: it cannot be split.
    4 = 2 × 2
    9 = 3 × 3
  3. Circle the primes at the ends of the branches. These are the prime factors.
    2, 2, 3, 3
  4. Write them in order, joined with ×. The smallest prime goes first.
    36 = 2 × 2 × 3 × 3

Check. Multiply the primes back together: 2 × 2 = 4, 4 × 3 = 12, 12 × 3 = 36. ✓

Watch out. Every branch must end in a prime. 9 is not prime (3 × 3), and 1 is never used. You can start with any factor pair: the primes at the end are always the same.

Worked example 3.5.4
Write 90 as a product of prime factors.

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Split the number into factors with a factor tree. Keep going until every branch ends in a prime number. The primes at the ends are the prime factors.

  1. Split the number into a factor pair. Pick any factor pair you know from the times tables.
    90 = 2 × 45
    2335154590
  2. Keep splitting every number that is not prime. A number that is not prime can be split again. Stop on a prime: it cannot be split.
    45 = 3 × 15
    15 = 3 × 5
  3. Circle the primes at the ends of the branches. These are the prime factors.
    2, 3, 3, 5
  4. Write them in order, joined with ×. The smallest prime goes first.
    90 = 2 × 3 × 3 × 5

Check. Multiply the primes back together: 2 × 3 = 6, 6 × 3 = 18, 18 × 5 = 90. ✓

Watch out. Every branch must end in a prime. 9 is not prime (3 × 3), and 1 is never used. You can start with any factor pair: the primes at the end are always the same.

Worked example 3.5.5
Write 48 as a product of prime factors. Start the tree with 48 = 6 × 8.

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Split the number into factors with a factor tree. Keep going until every branch ends in a prime number. The primes at the ends are the prime factors.

  1. Split the number into a factor pair. Pick any factor pair you know from the times tables.
    48 = 6 × 8
  2. Keep splitting every number that is not prime. A number that is not prime can be split again. Stop on a prime: it cannot be split.
    6 = 2 × 3
    8 = 2 × 4
    4 = 2 × 2
  3. Circle the primes at the ends of the branches. These are the prime factors.
    2, 3, 2, 2, 2
  4. Write them in order, joined with ×. The smallest prime goes first.
    48 = 2 × 2 × 2 × 2 × 3

Check. Multiply the primes back together: 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16, 16 × 3 = 48. ✓

Watch out. Every branch must end in a prime. 9 is not prime (3 × 3), and 1 is never used. You can start with any factor pair: the primes at the end are always the same.

Worked example 3.5.6
Write 84 as a product of prime factors. Start the tree with 84 = 7 × 12.

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Split the number into factors with a factor tree. Keep going until every branch ends in a prime number. The primes at the ends are the prime factors.

  1. Split the number into a factor pair. Pick any factor pair you know from the times tables.
    84 = 7 × 12
  2. Keep splitting every number that is not prime. A number that is not prime can be split again. Stop on a prime: it cannot be split.
    12 = 2 × 6
    6 = 2 × 3
  3. Circle the primes at the ends of the branches. These are the prime factors.
    7, 2, 2, 3
  4. Write them in order, joined with ×. The smallest prime goes first.
    84 = 2 × 2 × 3 × 7

Check. Multiply the primes back together: 2 × 2 = 4, 4 × 3 = 12, 12 × 7 = 84. ✓

Watch out. Every branch must end in a prime. 9 is not prime (3 × 3), and 1 is never used. You can start with any factor pair: the primes at the end are always the same.

Worked example 3.5.7
Find the missing numbers in the factor tree for 60.
2??2?1060

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. In a factor tree, the two numbers on the branches multiply to make the number above them. Use this to find each missing number: multiply the two numbers below, or divide the number above.

  1. Look at what each branch shows. Each number is split into a factor pair.
    Number above = number on the left × number on the right
  2. Find the missing numbers. Use multiplying or dividing, whichever you can do.
    The top number 60 is split into ? × 10. So ? = 60 ÷ 10 = 6.
    The top number 6 is split into ? × 2. So ? = 6 ÷ 2 = 3.
    The top number 10 is split into ? × 2. So ? = 10 ÷ 2 = 5.
  3. Write the missing numbers. List them in the order of the boxes (top to bottom, left to right).
    6, 3, 5

Check. Multiply the circled primes at the bottom: 2 × 2 × 3 × 5 = 60. ✓

Watch out. Every number at the end of a branch must be prime.

Worked example 3.5.8
A number is 2 × 2 × 5 × 5. What is the number?

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Multiply the prime factors together, two at a time, from left to right.

  1. Write the product. These are the prime factors of the number.
    2 × 2 × 5 × 5
  2. Multiply two at a time. Use the answer to each step in the next step.
    2 × 2 = 4
    4 × 5 = 20
    20 × 5 = 100
  3. Write the number.
    2 × 2 × 5 × 5 = 100

Check. Go backwards and divide 100 by each prime in turn: 100 ÷ 2 = 50, 50 ÷ 2 = 25, 25 ÷ 5 = 5, 5 ÷ 5 = 1. We reach 1. ✓

Watch out. Multiply the numbers. Do not add them.

Worked example 3.5.9
What are the prime factors of 72? Write each different prime once.

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Split the number into factors with a factor tree. Keep going until every branch ends in a prime number. The primes at the ends are the prime factors.

  1. Split the number into a factor pair. Pick any factor pair you know from the times tables.
    72 = 2 × 36
  2. Keep splitting every number that is not prime. A number that is not prime can be split again. Stop on a prime: it cannot be split.
    36 = 2 × 18
    18 = 2 × 9
    9 = 3 × 3
  3. Circle the primes at the ends of the branches. These are the prime factors.
    2, 2, 2, 3, 3
  4. List each different prime once. The question asks for the prime factors, not the whole product.
    72 = 2 × 2 × 2 × 3 × 3, so the prime factors are 2 and 3

Check. Multiply the primes back together: 2 × 2 = 4, 4 × 2 = 8, 8 × 3 = 24, 24 × 3 = 72. ✓

Watch out. Every branch must end in a prime. 9 is not prime (3 × 3), and 1 is never used. You can start with any factor pair: the primes at the end are always the same.

Watch out

Check that every number at the end of a branch is prime. 4, 6, 8, 9 and 10 are not prime, so they must be split again. Never use 1 in a factor tree (1 × 12 does not help). And do not leave out repeats: 12 = 2 × 2 × 3, not 2 × 3.

Exercise 3F  Prime factors and factor trees

Non-calculator
1
FWrite the missing numbers in each factor tree.
(a)
22?520
(b)
32??18
(c)
53??45
2
FUse a factor tree to write each number as a product of two prime numbers.
(a)14
(b)21
(c)35
(d)15
3
FWrite each number as a product of prime factors. Draw a factor tree to help.
(a)24
(b)40
(c)45
(d)50
4
FWrite each number as a product of prime factors.
(a)60
(b)63
(c)72
(d)80
5
FWrite each number as a product of prime factors. Take care: there are more than three primes in each.
(a)90
(b)96
(c)98
6
FEach number is written as a product of prime factors. Work out the number.
(a)2 × 2 × 3
(b)2 × 3 × 5
(c)3 × 3 × 5
(d)2 × 2 × 2 × 7
7
FWhat are the prime factors of each number? Write each different prime once.
(a)36
(b)100
(c)75
(d)84
8
PSSam and Mia each draw a factor tree for 48. Sam starts with 6 × 8. Mia starts with 4 × 12.
2362224848
22422361248
(a)Write the prime factors from Sam's tree as a product.
(b)Write the prime factors from Mia's tree as a product.
(c)What do you notice?
9
PSA number is 2 × 2 × 3 × 3. Is each of these a factor of the number?   Write Yes or No.
(a)12
(b)8
(c)18
(d)10
10
RNoah writes, “36 = 4 × 9, so the prime factors of 36 are 4 and 9.” Is Noah right? Explain.
11
RWhich number does not have 3 as a prime factor?   18    27    45    50
12
RAlways, sometimes or never?
(a)If a number is even, then 2 is one of its prime factors.
(b)If 3 is a prime factor of a number, then the number is odd.
13
RChallenge!   I am an odd number less than 100. I am made by multiplying exactly four prime numbers (they can be the same). What number am I?
3.6

Square numbers and cube numbers

Year 5
Square numbers and the notation for squared, Squares up to 12 × 12 and beyond, Cube numbers and the notation for cubed

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Square numbers and cube numbersComing soon

When you multiply a number by itself, you get a square number. 3 × 3 = 9, so 9 is a square number. If you arrange 9 dots or tiles in 3 rows of 3, they make a perfect square. We write 3 × 3 as 3² and say “3 squared”. The little 2 means “multiply the number by itself”. It does not mean “times 2”: 3² = 9, not 6.

1 × 1 = 12 × 2 = 43 × 3 = 94 × 4 = 165 × 5 = 25
The first five square numbers. Each new square is the last one plus an L-shape of dots: 1, then +3, +5, +7, +9. (1, 4, 9, 16, 25)

You should know the squares up to 12 × 12 straight away. They are in the table. Notice the pattern: you add the next odd number each time (1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16…).

Number123456789101112
Squared149162536496481100121144
Square numbers up to 12 × 12.

Squares beyond 12. Some bigger squares are very useful: 15² = 225, 20² = 400, 25² = 625 and 30² = 900. Here are three tricks.

Cube numbers. When you multiply a number by itself and then by itself again, you get a cube number. 2 × 2 × 2 = 8. We write it 2³ and say “2 cubed”. A cube number is the number of small cubes in a big cube. A 2 by 2 by 2 cube is made from 8 small cubes. A 3 by 3 by 3 cube has 27.

1 × 1 × 12 × 2 × 23 × 3 × 34 × 4 × 4
Cubes made from small cubes: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64.
Number12345678910
Cubed1827641252163435127291,000
Cube numbers up to 10³. Learn 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1,000.

A few numbers are both square and cube: 1, 64 (8² and 4³) and 729 (27² and 9³).

Key ideas

  • A square number is a number multiplied by itself: 6² = 6 × 6 = 36. The little 2 does not mean “times 2”.
  • Squares up to 12²: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Also know 15² = 225, 20² = 400, 25² = 625, 30² = 900.
  • A cube number is a number multiplied by itself twice: 4³ = 4 × 4 × 4 = 64. The little 3 does not mean “times 3”.
  • Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000.
  • To test if a number is square (or cube), list the squares (or cubes) and see if it is there.
Worked example 3.6.1
Work out 7².

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.

Method. 7² means 7 multiplied by itself: 7 × 7.

  1. Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.
    7² = 7 × 7
  2. Use a times table fact. You know this from the times tables.
    7 × 7 = 49
  3. Picture it. 7 rows of 7 dots make a square.
    7 rows × 7 columns = 49 dots

Check. The square numbers go up by the odd numbers: 36 + 13 = 49. ✓

Watch out. 7² is not 7 × 2 = 14. Multiply 7 by 7.

Worked example 3.6.2
Work out 15² in your head.

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.

Method. 15² means 15 multiplied by itself: 15 × 15.

  1. Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.
    15² = 15 × 15
  2. Split the second 15 into 10 and 5. Multiply 15 by each part. This is called partitioning.
    15 × 15 = 15 × 10 + 15 × 5
    15 × 10 = 150
    15 × 5 = 75
  3. Add the two answers.
    150 + 75 = 225

Check. Pattern for numbers ending in 5: 1 × 2 = 2, then write 25 on the end: 225. ✓

Watch out. 15² is not 15 × 2 = 30. Multiply 15 by 15.

Worked example 3.6.3
Work out 20².

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.

Method. 20² means 20 multiplied by itself: 20 × 20.

  1. Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.
    20² = 20 × 20
  2. Split each number into a single digit and a 10. 20 is 2 × 10. Multiplying can be done in any order.
    20 × 20 = (2 × 10) × (2 × 10) = (2 × 2) × (10 × 10)
  3. Multiply. 2 × 2 = 4 and 10 × 10 = 100.
    4 × 100 = 400

Check. 20 × 2 = 40, and 40 × 10 = 400. ✓

Watch out. 20² is not 20 × 2 = 40. Multiply 20 by 20.

Worked example 3.6.4
Work out 25².

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.

Method. 25² means 25 multiplied by itself: 25 × 25.

  1. Say what the little 2 means. The little 2 means “multiply the number by itself”. It does not mean “times 2”.
    25² = 25 × 25
  2. Split the second 25 into 20 and 5. Multiply 25 by each part. This is called partitioning.
    25 × 25 = 25 × 20 + 25 × 5
    25 × 20 = 500
    25 × 5 = 125
  3. Add the two answers.
    500 + 125 = 625

Check. Pattern for numbers ending in 5: 2 × 3 = 6, then write 25 on the end: 625. ✓

Watch out. 25² is not 25 × 2 = 50. Multiply 25 by 25.

Worked example 3.6.5
Is 50 a square number?

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.

Method. Write the square numbers in order, and see if 50 is one of them.

  1. List the square numbers. Keep going until you reach or pass 50.
    1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49, 8² = 64
  2. Look for the number.
    50 is between 49 and 64. It is not in the list.
  3. Answer the question.
    50 is not a square number: No

Check. No whole number multiplied by itself makes 50. ✓

Watch out. Do not guess from the last digit. Check the list of square numbers.

Worked example 3.6.6
Work out 4³.

Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.

Method. 4³ means 4 × 4 × 4. Multiply the first two, then multiply the answer by 4 again.

  1. Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.
    4³ = 4 × 4 × 4
  2. Multiply the first two numbers. Work from left to right.
    4 × 4 = 16
  3. Multiply that answer by 4 again.
    16 × 4 = 64

Check. Picture a cube: 4 layers, each with 16 small cubes. 4 × 16 = 64. ✓

Watch out. 4³ is not 4 × 3 = 12. It is 4 × 4 × 4.

Worked example 3.6.7
Work out 6³.

Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.

Method. 6³ means 6 × 6 × 6. Multiply the first two, then multiply the answer by 6 again.

  1. Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.
    6³ = 6 × 6 × 6
  2. Multiply the first two numbers. Work from left to right.
    6 × 6 = 36
  3. Multiply that answer by 6 again.
    36 × 6 = 30 × 6 + 6 × 6 = 180 + 36 = 216

Check. Picture a cube: 6 layers, each with 36 small cubes. 6 × 36 = 216. ✓

Watch out. 6³ is not 6 × 3 = 18. It is 6 × 6 × 6.

Worked example 3.6.8
Is 125 a cube number?

Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.

Method. Write the cube numbers in order, and see if 125 is one of them.

  1. List the cube numbers. Keep going until you reach or pass 125.
    1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125
  2. Look for the number.
    125 is in the list.
  3. Answer the question.
    125 is a cube number: Yes

Check. 5 × 5 × 5 = 125. ✓

Watch out. Do not guess from the last digit. Check the list of cube numbers.

Worked example 3.6.9
Is 64 a square number, a cube number, both or neither?

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.

Method. Check the square numbers, then check the cube numbers.

  1. Is it a square number? List squares until you reach or pass the number.
    1² = 1    2² = 4    3² = 9    4² = 16
    5² = 25    6² = 36    7² = 49    8² = 64
    9² = 81
    yes
  2. Is it a cube number? List cubes until you reach or pass the number.
    1³ = 1    2³ = 8    3³ = 27    4³ = 64
    5³ = 125
    yes
  3. Decide.
    64 is a square number and a cube number.

Check. 8 × 8 = 64. ✓

Watch out. A number can be both a square and a cube. 64 = 8 × 8 and 64 = 4 × 4 × 4.

Worked example 3.6.10
Work out 3² + 2³.

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.

Method. Work out each power first. Then do the addition or subtraction.

  1. Work out 3². 3² means 3 × 3.
    3² = 3 × 3 = 9
  2. Work out 2³. 2³ means 2 × 2 × 2.
    2³ = 2 × 2 × 2 = 8
  3. Now add. Use the two answers.
    9 + 8 = 17

Check. Estimate: 9 is about 10 and 8 is about 10, so the answer should be near 20. ✓

Watch out. Do not multiply the number by the little number: 3² is not 3 × 2.

Watch out

3² is not 3 × 2 and 3³ is not 3 × 3. 3² = 3 × 3 = 9, and 3³ = 3 × 3 × 3 = 27. Also, do not mix up squares and cubes: 4² = 16 but 4³ = 64. And remember that 25² is not 50: it is 25 × 25 = 625.

Exercise 3G  Square numbers

Non-calculator
1
FThe picture shows the first five square numbers.
1 × 1 = 12 × 2 = 43 × 3 = 94 × 4 = 165 × 5 = 25
(a)How many dots are in the 6th square?
(b)How many more dots does the 6th square have than the 5th square?
(c)Write the 6th square number using the notation for squared.
2
FWrite each one as a multiplication. Then work it out.
(a)3²
(b)5²
(c)8²
(d)11²
(e)12²
3
FFind the missing number.   What number multiplied by itself makes…
(a)?² = 49
(b)?² = 100
(c)?² = 144
(d)?² = 81
4
FWork out each square. You can use partitioning or a trick.
(a)13²
(b)15²
(c)20²
(d)25²
5
FWork out each square. Think about 3 × 3 = 9 and 10 × 10 = 100.
(a)30²
(b)40²
(c)50²
(d)80²
6
FIs each number a square number?   Write Yes or No.
(a)36
(b)50
(c)81
(d)90
(e)121
(f)200
7
FWrite the missing numbers in this list of square numbers.
1, 4, ?, 16, ?, 36, 49, ?, 81, ?
8
PSSolve each problem.
(a)A square patio has 13 slabs along each side. How many slabs are there altogether?
(b)A square tray holds 144 chocolates in equal rows. How many chocolates are in each row?
9
RLeo says, “6² = 12.” Is Leo right? Explain his mistake.
10
RAlways, sometimes or never?
(a)The sum of two odd numbers is an even number.
(b)A square number is a multiple of 4.
(c)A square number ends in the digit 7.
11
RProve it!   Show that 3² + 4² = 5².
12
RChallenge!   I am a square number between 20 and 80. I am an even number. I am a multiple of 3. What number am I?

Exercise 3H  Cube numbers

Non-calculator
1
FThe pictures show cubes made from small cubes.
1 × 1 × 12 × 2 × 23 × 3 × 34 × 4 × 4
(a)How many small cubes are in the 1 × 1 × 1 cube?
(b)How many small cubes are in the 2 × 2 × 2 cube?
(c)How many small cubes are in the 3 × 3 × 3 cube?
(d)How many small cubes are in the 4 × 4 × 4 cube?
2
FWork out each cube number.
(a)2³
(b)3³
(c)5³
(d)10³
3
FWrite each multiplication using the notation for cubed. Then work it out.
(a)6 × 6 × 6
(b)7 × 7 × 7
(c)9 × 9 × 9
4
FIs each number a cube number?   Write Yes or No.
(a)8
(b)20
(c)125
(d)100
(e)343
(f)1,000
5
FWrite the missing numbers in this list of cube numbers.
1, 8, ?, 64, ?, 216, ?, 512
6
FWork out each one. Do the squares and cubes first.
(a)5² − 2³
(b)4³ − 6²
(c)2³ + 4²
(d)10³ − 10²
7
RMia says, “4³ is bigger than 8².” Is Mia right? Show how you know.
8
PSA 3 by 3 by 3 cube is made from small cubes. The outside of the big cube is painted blue.
(a)How many small cubes make the big cube?
(b)How many of the small cubes have blue paint on at least one face?
9
PSSolve each problem.
(a)A cube-shaped box is filled with sugar cubes. There are 10 sugar cubes along each edge. How many sugar cubes are there?
(b)A cube-shaped box holds exactly 125 small cubes. How many small cubes are along each edge?
10
RSome numbers are both square numbers and cube numbers.
(a)Find a number between 50 and 100 that is both a square and a cube.
(b)Find a number between 700 and 800 that is both a square and a cube.
3.7

Putting it all together: clues and reasoning

Year 5
Problems linking multiples, factors, primes, squares and cubes, Number clues, Reasoning: always, sometimes, never; prove it; spot the mistake

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Putting it all together: clues and reasoningComing soon

In this section you use everything from the chapter. Many problems give clues about a mystery number. Here is a good way to solve them.

  1. Read all the clues. Choose the clue that gives the shortest list, for example “a square number” or “a prime number”.
  2. Write the list of numbers that fit this clue (inside the range).
  3. Use the other clues, one at a time, to cross out numbers.
  4. Check the last number left against every clue.

I am a square numberI am a multiple of 3I am between 30 and 70

The squares between 30 and 70 are 36, 49 and 64. Only 36 is a multiple of 3. So the number is 36.

How many factors? Here is a hundred-square style table of the numbers from 1 to 50. Numbers with exactly 2 factors (the primes) are gold. Numbers with exactly 3 factors are green. Numbers with exactly 4 factors are blue. Can you see a pattern? The green numbers are 4, 9, 25 and 49. They are the squares of prime numbers (2², 3², 5², 7²). The number with the most factors under 50 is 48. It has 10: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

12345678910
11121314151617181920
21222324252627282930
31323334353637383940
41424344454647484950
Gold: exactly 2 factors (prime). Green: exactly 3 factors. Blue: exactly 4 factors. 1 is purple (1 factor).

Reasoning. The big reasoning questions are about what is always, sometimes or never true. To show “sometimes” you need one example that works and one that does not. An example that breaks a rule is a counter-example. To prove something is always true, explain why in words: examples alone are not enough. To spot a mistake, work the question out yourself first, then compare.

Key ideas

  • For a clue puzzle, start with the clue that gives the shortest list. Cross out numbers with each other clue. Check the answer against every clue.
  • Numbers with exactly 2 factors are prime. Numbers with exactly 3 factors are the squares of primes (4, 9, 25, 49).
  • To show “sometimes” give one example that works and one that does not. One counter-example shows “always” is false.
  • A proof needs a reason that works for every number, not just a few examples.
  • For a problem about things that repeat (buses, lights, bells), use common multiples. For sharing into equal groups, use common factors.
Worked example 3.7.1
I am a square number. I am a multiple of 3. I am between 30 and 70. What number am I?

Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.

Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.

  1. Start with the clue that gives a short list: “a square number”. List the numbers between 30 and 70 that fit this clue.
    36, 49, 64
  2. Use the next clue: “a multiple of 3”. Cross out every number that does not fit.
    36, 49, 64
  3. Check the answer against every clue. It must fit all of them.
    “a square number” ✓
    “a multiple of 3” ✓

Check. Only 36 is left, and it fits every clue. ✓

Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.

Worked example 3.7.2
I am a prime number between 20 and 30. My digits add up to 11. What number am I?

Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.

Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.

  1. Start with the clue that gives a short list: “a prime number”. List the numbers between 20 and 30 that fit this clue.
    23, 29
  2. Use the next clue: “my digits add up to 11”. Cross out every number that does not fit.
    23, 29
  3. Check the answer against every clue. It must fit all of them.
    “a prime number” ✓
    “my digits add up to 11” ✓

Check. Only 29 is left, and it fits every clue. ✓

Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.

Worked example 3.7.3
I am a common multiple of 4 and 6. I am a multiple of 9. I am less than 40. What number am I?

Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.

Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.

  1. Start with the clue that gives a short list: “a multiple of 9”. List the numbers less than 40 that fit this clue.
    9, 18, 27, 36
  2. Use the next clue: “a multiple of 4”. Cross out every number that does not fit.
    9, 18, 27, 36
  3. Use the next clue: “a multiple of 6”. Cross out every number that does not fit.
    36
  4. Check the answer against every clue. It must fit all of them.
    “a multiple of 9” ✓
    “a multiple of 4” ✓
    “a multiple of 6” ✓

Check. Only 36 is left, and it fits every clue. ✓

Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.

Worked example 3.7.4
Which numbers less than 50 have exactly three factors?

Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

Method. Count the factors of each number. Write down the ones with exactly 3 factors.

  1. Find how many factors each number has. Use factor pairs. The list shows number:count.
    2:2   3:2   4:3   5:2   6:4   7:2   8:4   9:3   10:4   11:2
    12:6   13:2   14:4   15:4   16:5   17:2   18:6   19:2   20:6   21:4
    22:4   23:2   24:8   25:3   26:4   27:4   28:6   29:2   30:8   31:2
    32:6   33:4   34:4   35:4   36:9   37:2   38:4   39:4   40:8   41:2
    42:8   43:2   44:6   45:6   46:4   47:2   48:10   49:3
  2. Pick the numbers with exactly 3 factors. They are in bold.
    4, 9, 25, 49
  3. Look for a pattern.
    Each of these is the square of a prime number (2², 3², 5², …).

Check. Check one: 49 has factors 1, 7, 49, which is 3. ✓

Watch out. Count 1 and the number itself too.

Worked example 3.7.5
Which number less than 50 has the most factors?

Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

Method. Count the factors of every number in the range. To save time, notice that numbers with lots of factors are multiples of small numbers like 6 or 12. Make a table of the ones with the most.

  1. Count the factors of each number. Use factor pairs. Most numbers have only a few factors.
    Every number from 1 to 49 that is not in the table below has 6 factors or fewer.
  2. List the numbers with 8 or more factors. These are the only ones that could win.
    NumberFactorsHow many
    241, 2, 3, 4, 6, 8, 12, 248
    301, 2, 3, 5, 6, 10, 15, 308
    361, 2, 3, 4, 6, 9, 12, 18, 369
    401, 2, 4, 5, 8, 10, 20, 408
    421, 2, 3, 6, 7, 14, 21, 428
    481, 2, 3, 4, 6, 8, 12, 16, 24, 4810
  3. Find the most. Look at the last column.
    The most is 10 factors: 48

Check. Check one: 48 = 1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8, which gives 10 factors. ✓

Watch out. Count 1 and the number itself. Do not stop at the first few pairs.

Worked example 3.7.6
Three bells ring together at noon. Bell A rings every 4 minutes, bell B every 6 minutes and bell C every 10 minutes. After how many minutes do they next ring together?

Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.

Method. Write the multiples of the biggest number. Test each one against the other numbers. The first one that is in every times table is the LCM.

  1. List the multiples of 10. The common multiple must be in the times table of the biggest number, so we only test these.
    10, 20, 30, 40, 50, 60
  2. Test each one. A number passes if it divides exactly by 4 and by 6.
    10: not in the 4 times table (10 ÷ 4 = 2 remainder 2)
    20: not in the 6 times table (20 ÷ 6 = 3 remainder 2)
    30: not in the 4 times table (30 ÷ 4 = 7 remainder 2)
    40: not in the 6 times table (40 ÷ 6 = 6 remainder 4)
    50: not in the 4 times table (50 ÷ 4 = 12 remainder 2)
    60: in all the tables
  3. Write the answer. We stop at the first number that passes every test.
    The LCM of 4, 6 and 10 is 60.

Check. 60 ÷ 4 = 15 , 60 ÷ 6 = 10 , 60 ÷ 10 = 6. All exact. ✓

Watch out. Test every number, not just one of them.

Worked example 3.7.7
Prove it!   Every multiple of 6 is also a multiple of 3.

Key words. A counter-example is one example that shows a statement is false.

Method. Try some examples. If the statement is true for every example you can find, ask why. If you find one that is false, it cannot be “always”.

  1. Try some numbers. Test the statement with some examples.
    6 = 3 × 2, so 6 is in the 3 times table. Multiples of 6: 6, 12, 18, 24, 30. Each is also in the 3 times table: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
  2. Explain why. A reason shows what is happening for every number, not just the ones we tried.
    6 = 3 × 2. A multiple of 6 is 6 × (any number) = 3 × 2 × (any number) = 3 × (something). So it is always in the 3 times table.
  3. Decide.
    Always

Check. 42 = 6 × 7 = 3 × 14. ✓

Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not. “Never” means no example works.

Worked example 3.7.8
Jack says, “The lowest common multiple of 4 and 10 is 40.” Is Jack right?

Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.

Method. Work out the correct answer yourself first. Then compare it with the answer in the question to find the mistake.

  1. Work it out carefully. Do the question again, one step at a time.
    Multiples of 4: 4, 8, 12, 16, 20
    Multiples of 10: 10, 20
    The first number in both lists is 20.
  2. Compare with the answer in the question.
    Jack said 40.
  3. Find the mistake.
    40 is a common multiple (4 × 10 = 40), but it is not the lowest. 20 is in both lists and it is smaller.
  4. Write the correct answer.
    No. The LCM of 4 and 10 is 20.

Check. 20 ÷ 4 = 5 and 20 ÷ 10 = 2. ✓

Watch out. Multiplying the two numbers does not always give the lowest common multiple.

Worked example 3.7.9
Which number is the odd one out?   16    25    36    48

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.

Method. Look for something three numbers share. Try square numbers.

  1. Write each as a multiplication. See whether it is a number multiplied by itself.
    16 = 4 × 4
    25 = 5 × 5
    36 = 6 × 6
    48 = 6 × 8 (not equal numbers)
  2. Decide. 16, 25 and 36 are square numbers. 48 is not.
    48 is the odd one out.
  3. Check the squares. Make sure 48 is not a square: 6² = 36 and 7² = 49.
    48 is between 36 and 49, so it is not a square.

Check. The squares are 1, 4, 9, 16, 25, 36, 49, … and 48 is not in the list. ✓

Watch out. There may be another reason too (25 is the only odd number here). Say which reason you use. Both can be right.

Watch out

Do not stop when you find a number that fits the first clue. Check it against all the clues. And in an “always, sometimes, never” question, one example is never enough to prove “always”. Remember “between 30 and 70” does not include 30 or 70.

Exercise 3I  Number clues and reasoning

Non-calculator
1
FI am a multiple of 7 between 50 and 80. I am an odd number. My digits add up to 9. What number am I?
2
FI am a factor of 36. I am an even number. I am a multiple of 3. I am between 8 and 15. What number am I?
3
FI am a prime number between 50 and 70. My digits add up to 8. What number am I?
4
PSI am a square number between 100 and 1,000. I am a multiple of 5 and a multiple of 6. What number am I?
5
FFind all the numbers with exactly…
(a)2 factors, up to 20
(b)3 factors, up to 30
(c)4 factors, up to 20
6
PSWhich numbers between 50 and 100 have the most factors? How many factors is that?
7
PSOne lighthouse flashes every 6 seconds. Another lighthouse flashes every 9 seconds. They flash together now (at 0 seconds).
(a)After how many seconds do they next flash together?
(b)How many times do they flash together in the next 2 minutes (120 seconds), not counting the start?
8
PSThree buses leave a stop at 8:00 am. Bus 1 leaves every 3 minutes, bus 2 every 5 minutes and bus 3 every 6 minutes. After how many minutes do all three buses leave together again?
9
RProve it!   Show that 93 is not a prime number.
10
RAlways, sometimes or never?
(a)A multiple of 4 is a multiple of 8.
(b)A factor of 12 is a factor of 24.
(c)A cube number is an even number.
11
RWhich number is the odd one out? Explain.   16    36    64    50
12
RZainab says, “The lowest common multiple of 5 and 6 is 11, because 5 + 6 = 11.” Is Zainab right? Explain.
13
RChallenge!   Find the number less than 50 that has exactly five factors.

Chapter summary

SkillMethod
MultiplesMultiply the number by 1, 2, 3, 4… (count on in its steps). To test a number, look at the multiples just below and just above it, or divide: no remainder means it is a multiple.
Common multiples and LCMList the multiples of both numbers. The numbers in both lists are common multiples. The smallest is the lowest common multiple (LCM). Use it for buses, lights and bells that repeat.
Tests of divisibility2: last digit even. 3: digits add to a multiple of 3. 4: last two digits in the 4s (or halve twice). 5: ends in 0 or 5. 6: passes 2 and 3. 8: halve three times. 9: digits add to a multiple of 9. 10: ends in 0. 25: ends in 00, 25, 50, 75. 100: ends in 00.
Factors and factor pairsStart with 1 × the number. Try 2, 3, 4… in order and write each pair that divides exactly. Stop when the pairs meet. Write the factors in order. Square numbers have an odd number of factors.
Common factors and HCFList the factors of both numbers. The numbers in both lists are the common factors. The biggest is the highest common factor (HCF). Use it for sharing into equal groups with none left over.
Prime and composite numbersPrime: exactly two factors (1 and itself). Composite: more than two factors. 1 is neither. 2 is the only even prime. Primes to 19: 2, 3, 5, 7, 11, 13, 17, 19.
Is a number up to 100 prime?Test with 2, 3, 5 and 7. If one divides it (and the number is not that prime itself), it is composite. If none does, it is prime. The sieve crosses out the multiples of 2, 3, 5 and 7.
Prime factors and factor treesSplit the number into a factor pair, split again, and stop on primes. Write the primes in order with × between them: 60 = 2 × 2 × 3 × 5. Check by multiplying.
Square numbersn² = n × n. Know 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and 15² = 225, 20² = 400, 25² = 625, 30² = 900. The little 2 does not mean “times 2”.
Cube numbersn³ = n × n × n. Know 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000. The little 3 does not mean “times 3”.
Clues and reasoningFor a clue puzzle, start with the clue that gives the shortest list, cross out, then check every clue. “Always” needs a reason; “sometimes” needs two examples; one counter-example breaks “always”.

Can you…?

  • I can list multiples of a number and decide whether a number is a multiple of another.
  • I can find common multiples and the lowest common multiple, and use them to solve problems about things that repeat.
  • I can use the tests of divisibility for 2, 3, 4, 5, 6, 8, 9, 10, 25 and 100.
  • I can find all the factor pairs of a number in order, and list its factors.
  • I can find common factors and the highest common factor of two numbers.
  • I know what prime and composite numbers are, and I know the primes up to 19.
  • I can decide whether a number up to 100 is prime.
  • I can write a number up to 100 as a product of prime factors using a factor tree.
  • I can work out squares (up to 12 × 12 and beyond) and cubes, and use the ² and ³ notation.
  • I can solve clue puzzles and explain my reasoning: always, sometimes, never; prove it; spot the mistake.

Key terms

Multiple
A number in the times table of another number. The multiples of 5 are 5, 10, 15, 20…
Common multiple
A number that is a multiple of two (or more) numbers.
Lowest common multiple (LCM)
The smallest common multiple. The LCM of 4 and 6 is 12.
Divisible
A number is divisible by another if it divides exactly, with no remainder.
Test of divisibility
A quick rule, using the digits, to find out whether a number is divisible by another.
Factor
A whole number that divides exactly into another number. The factors of 6 are 1, 2, 3 and 6.
Factor pair
Two factors that multiply to make a number: 2 and 3 are a factor pair of 6.
Common factor
A number that is a factor of two (or more) numbers.
Highest common factor (HCF)
The biggest common factor. The HCF of 12 and 18 is 6.
Prime number
A number with exactly two factors: 1 and itself. 2, 3, 5, 7, 11…
Composite number
A number with more than two factors, such as 12. (1 is neither prime nor composite.)
Prime factor
A factor that is a prime number. The prime factors of 12 are 2 and 3.
Factor tree
A diagram that splits a number into factors until every branch ends in a prime number.
Product
The answer to a multiplication. 2 × 2 × 3 is a product of primes.
Square number
A number multiplied by itself, such as 25 = 5 × 5. Written 5².
Cube number
A number multiplied by itself twice, such as 125 = 5 × 5 × 5. Written 5³.
Sieve of Eratosthenes
A way of finding primes on a hundred square by crossing out the multiples of 2, 3, 5 and 7.
Counter-example
One example that shows a statement is false.
Array
Objects arranged in equal rows and columns. It shows a multiplication and a factor pair.

Exercise 3Z  Check-up (non-calculator)

Non-calculator
1
Write the first five multiples of 9.
2
Find the lowest common multiple (LCM) of 6 and 10.
3
Is 3,456 divisible by 8?   Write Yes or No.
4
Write all the factors of 40.
5
Find the highest common factor (HCF) of 24 and 40.
6
Is 83 a prime number?   Write Prime or Not prime.
7
Write 56 as a product of prime factors.
8
Work out 8² + 3³.
9
Find the missing number.   ?³ = 216
10
I am a multiple of 8. I am a multiple of 3. I am between 60 and 80. What number am I?

Total 18 marks. Allow about 18 minutes.

Exercise 3X  SATs-style questions

Non-calculator
1
FWhich number is a multiple of both 4 and 6?   Tick one box.   A 16   B 18   C 24   D 28[1 mark]
2
FCircle the prime number.   21    27    29    33[1 mark]
3
FWhat is the value of 6³?[1 mark]
4
FWhich number is divisible by 9?   Tick one box.   A 4,321   B 5,076   C 6,125   D 7,890[1 mark]
5
RJack says, “77 is a prime number.” Show that Jack is wrong.[2 marks]
6
FWrite all the factors of 28.[2 marks]
7
PSThe red bus leaves the station every 8 minutes. The blue bus leaves every 12 minutes. Both buses leave at 10:00 am. At what time do they next leave together?   Show your working.[2 marks]
Red81624Blue1224minutes after 10:00 am
8
FWrite 72 as a product of prime factors.   Draw a factor tree to help you.[2 marks]
9
PSTara has 24 red beads and 36 blue beads. She makes necklaces that are all the same. She uses every bead.[3 marks]
(a)What is the greatest number of necklaces she can make? (2 marks)
(b)How many red beads and how many blue beads are on each necklace? (1 mark)
10
RDev thinks of a number. It is a square number. It is also a cube number. It is more than 1 and less than 100. What is Dev’s number?   Show how you know.[3 marks]
← Chapter 2: Adding and subtractingChapter 4: Multiplication →

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