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Primary Maths · Year 4Chapter 3 · Times tables to 12 × 12
Year 4 · Times Tables · Lesson 3.6

Year 4: Factor pairs and multiplying three numbers

Part of the Year 4 times tables chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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3.6

Factor pairs and multiplying three numbers

Year 4
Factor pairs, Multiplying three numbers

A factor of a number is a whole number that divides into it exactly, with nothing left over. A factor pair is two numbers that multiply to make the number. The factor pairs of 12 are 1 × 12, 2 × 6 and 3 × 4. Each pair makes a different array of 12 squares.

1 × 122 × 63 × 4
The three factor pairs of 12 as arrays. The factors of 12 are 1, 2, 3, 4, 6 and 12.

How to find all the factor pairs. Start at 1 and test each number in turn using the times tables: does it fit into the number exactly? Write down every pair you find. You can stop when the numbers meet in the middle, because after that you would only find the same pairs the other way round. Then write the factors in order. Every number has the pair 1 and itself.

Square numbers have a pair with the same number twice, like 6 × 6 = 36. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. Notice that 6 is only written once.

Factors and multiples. Factors are smaller than (or equal to) the number. Multiples are bigger than (or equal to) the number. The factors of 12 are 1, 2, 3, 4, 6, 12. The multiples of 12 are 12, 24, 36, 48...

Multiplying three numbers. Suppose a baker has 2 trays. Each tray has 3 rows of 4 buns. The number of buns is 2 × 3 × 4. You can multiply in any order and the answer is the same: (2 × 3) × 4 = 6 × 4 = 24, or 2 × (3 × 4) = 2 × 12 = 24. So look for a pair that is easy to multiply first. A great pair to look for is one that makes 10: 2 × 5. For 2 × 5 × 7, do 2 × 5 = 10, then 10 × 7 = 70.

2 trays with 3 rows of 4 buns: 2 × 3 × 4 = 24.

Key words. A factor divides into a number exactly. A factor pair is two factors that multiply to make the number. A multiple is in the times table of a number.

Key ideas

  • A factor pair is two numbers that multiply to make the number: 3 × 4 = 12.
  • Test 1, 2, 3, ... in order. Stop when the numbers meet in the middle.
  • Every number has 1 and itself as a factor pair.
  • You can multiply three numbers in any order. Look for a pair that makes 10.
  • Factors are small and multiples are big: factors of 12: 1, 2, 3, 4, 6, 12. Multiples of 12: 12, 24, 36...
Worked example 3.6.1
Find all the factor pairs of 12.

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.

Method. A factor pair is two numbers that multiply to make 12. Test 1, 2, 3, ... in order, and write down every pair you find.

  1. Start at 1 and try each number in turn. A factor pair is two numbers that multiply to make the number. We test 1, 2, 3, ... using the times tables.
    1 × 12 = 12  (a pair: 1 and 12)
    2 × 6 = 12  (a pair: 2 and 6)
    3 × 4 = 12  (a pair: 3 and 4)
  2. Stop when the numbers meet. After 3, the next factors would just be pairs we have already found the other way round.
    We can stop at 3.
  3. Write the factor pairs.
    1 × 12 = 12
    2 × 6 = 12
    3 × 4 = 12
  4. Picture them. Each pair makes a different array of the same number of squares.
    1 × 122 × 63 × 4
  5. List all the factors in order. Each number in a pair is a factor.
    1, 2, 3, 4, 6, 12

Check. There are 6 factors: 1, 2, 3, 4, 6, 12. Every factor pairs up with another. ✓

Watch out. Do not forget 1 and the number itself. Every number has the pair 1 and itself.

Worked example 3.6.2
Find all the factor pairs of 24.

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.

Method. A factor pair is two numbers that multiply to make 24. Test 1, 2, 3, ... in order, and write down every pair you find.

  1. Start at 1 and try each number in turn. A factor pair is two numbers that multiply to make the number. We test 1, 2, 3, ... using the times tables.
    1 × 24 = 24  (a pair: 1 and 24)
    2 × 12 = 24  (a pair: 2 and 12)
    3 × 8 = 24  (a pair: 3 and 8)
    4 × 6 = 24  (a pair: 4 and 6)
  2. Stop when the numbers meet. After 4, the next factors would just be pairs we have already found the other way round.
    We can stop at 4.
  3. Write the factor pairs.
    1 × 24 = 24
    2 × 12 = 24
    3 × 8 = 24
    4 × 6 = 24
  4. Picture them. Each pair makes a different array of the same number of squares.
    1 × 242 × 123 × 84 × 6
  5. List all the factors in order. Each number in a pair is a factor.
    1, 2, 3, 4, 6, 8, 12, 24

Check. There are 8 factors: 1, 2, 3, 4, 6, 8, 12, 24. Every factor pairs up with another. ✓

Watch out. Do not forget 1 and the number itself. Every number has the pair 1 and itself.

Worked example 3.6.3
Find all the factor pairs of 36.

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.

Method. A factor pair is two numbers that multiply to make 36. Test 1, 2, 3, ... in order, and write down every pair you find.

  1. Start at 1 and try each number in turn. A factor pair is two numbers that multiply to make the number. We test 1, 2, 3, ... using the times tables.
    1 × 36 = 36  (a pair: 1 and 36)
    2 × 18 = 36  (a pair: 2 and 18)
    3 × 12 = 36  (a pair: 3 and 12)
    4 × 9 = 36  (a pair: 4 and 9)
    5 × 7 = 35 and 5 × 8 = 40: 36 is in between, so 5 is not a factor
    6 × 6 = 36  (6 is paired with itself)
  2. Stop when the numbers meet. After 6, the next factors would just be pairs we have already found the other way round.
    We can stop at 6.
  3. Write the factor pairs.
    1 × 36 = 36
    2 × 18 = 36
    3 × 12 = 36
    4 × 9 = 36
    6 × 6 = 36
  4. Picture them. Each pair makes a different array of the same number of squares.
    1 × 362 × 183 × 124 × 96 × 6
  5. List all the factors in order. Each number in a pair is a factor.
    1, 2, 3, 4, 6, 9, 12, 18, 36

Check. There are 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, 36. Every factor pairs up with another. ✓

Watch out. Do not forget 1 and the number itself. Every number has the pair 1 and itself.

Worked example 3.6.4
Find all the factor pairs 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.

Method. A factor pair is two numbers that multiply to make 30. Test 1, 2, 3, ... in order, and write down every pair you find.

  1. Start at 1 and try each number in turn. A factor pair is two numbers that multiply to make the number. We test 1, 2, 3, ... using the times tables.
    1 × 30 = 30  (a pair: 1 and 30)
    2 × 15 = 30  (a pair: 2 and 15)
    3 × 10 = 30  (a pair: 3 and 10)
    4 × 7 = 28 and 4 × 8 = 32: 30 is in between, so 4 is not a factor
    5 × 6 = 30  (a pair: 5 and 6)
  2. Stop when the numbers meet. After 5, the next factors would just be pairs we have already found the other way round.
    We can stop at 5.
  3. Write the factor pairs.
    1 × 30 = 30
    2 × 15 = 30
    3 × 10 = 30
    5 × 6 = 30
  4. Picture them. Each pair makes a different array of the same number of squares.
    1 × 302 × 153 × 105 × 6
  5. List all the factors in order. Each number in a pair is a factor.
    1, 2, 3, 5, 6, 10, 15, 30

Check. There are 8 factors: 1, 2, 3, 5, 6, 10, 15, 30. Every factor pairs up with another. ✓

Watch out. Do not forget 1 and the number itself. Every number has the pair 1 and itself.

Worked example 3.6.5
Is 7 a factor of 42?

Method. Check if 42 is in the 7 times table. If it is, 7 is a factor.

  1. Say what we need to check. A factor divides into the number exactly, with nothing left over. So 7 is a factor of 42 only if 42 is in the 7 times table.
    Is 42 in the 7 times table?
  2. Count up in 7s. Find the multiples of 7 close to 42.
    7 × 6 = 42
  3. Answer.
    Yes. 7 × 6 = 42, so 7 is a factor of 42.

Check. 42 ÷ 7 = 6 exactly, with nothing left over. ✓

Watch out. A factor must divide in exactly, with nothing left over.

Worked example 3.6.6
Is 5 a factor of 42?

Method. Check if 42 is in the 5 times table. If it is, 5 is a factor.

  1. Say what we need to check. A factor divides into the number exactly, with nothing left over. So 5 is a factor of 42 only if 42 is in the 5 times table.
    Is 42 in the 5 times table?
  2. Count up in 5s. Find the multiples of 5 on each side of 42.
    5 × 8 = 40
    5 × 9 = 45
  3. Answer. 42 is between those two multiples, so it is not in the 5 times table.
    No. 5 is not a factor of 42.

Check. 42 ÷ 5 = 8 remainder 2. There is something left over. ✓

Watch out. A factor must divide in exactly, with nothing left over.

Worked example 3.6.7
Work out 2 × 3 × 4.

Key words. You can multiply three numbers in any order and the answer is the same.

Method. Multiplication can be done in any order, so look for two numbers that are easy to multiply first.

  1. Choose two numbers to multiply first. 2 × 3 is a times table fact.
    2 × 3 = 6
  2. Multiply the answer by the last number, 4. 6 × 4 is a times table fact.
    6 + 6 = 12
    double 10 = 20, double 2 = 4, so 20 + 4 = 24
    6 × 4 = 24
  3. Write the answer.
    2 × 3 × 4 = 24

Check. Try a different order: 3 × 4 = 12, then 12 × 2 = 24. ✓

Watch out. You can multiply in any order. Pick the pair that gives an easy answer, like a pair that makes 10.

Worked example 3.6.8
Work out 2 × 5 × 7.

Key words. You can multiply three numbers in any order and the answer is the same.

Method. Multiplication can be done in any order, so look for two numbers that are easy to multiply first.

  1. Choose two numbers to multiply first. 2 × 5 = 10, and multiplying by 10 is easy.
    2 × 5 = 10
  2. Multiply the answer by the last number, 7. 10 × 7 is a times table fact.
    7 tens
    7 → 70
    10 × 7 = 70
  3. Write the answer.
    2 × 5 × 7 = 70

Check. Try a different order: 5 × 7 = 35, then 35 × 2 = 70. ✓

Watch out. You can multiply in any order. Pick the pair that gives an easy answer, like a pair that makes 10.

Worked example 3.6.9
Work out 4 × 3 × 5.

Key words. You can multiply three numbers in any order and the answer is the same.

Method. Multiplication can be done in any order, so look for two numbers that are easy to multiply first.

  1. Choose two numbers to multiply first. 4 × 5 = 20, a multiple of 10, which is easy to multiply.
    4 × 5 = 20
  2. Multiply the answer by the last number, 3. 20 is 2 tens. Work out 2 × 3, then say that many tens.
    2 × 3 = 6, so 2 tens × 3 = 6 tens
    6 tens = 60
  3. Write the answer.
    4 × 3 × 5 = 60

Check. Try a different order: 5 × 3 = 15, then 15 × 4 = 60. ✓

Watch out. You can multiply in any order. Pick the pair that gives an easy answer, like a pair that makes 10.

Watch out

Do not forget 1 and the number itself! They are always a factor pair. Also, a factor pair must multiply to make the number exactly: 3 × 5 = 15, but 3 and 5 are not a factor pair of 16. When you multiply three numbers, only multiply two at a time. Do the first pair, then multiply the answer by the third number.

Exercise 3F  Factor pairs and multiplying three numbers

Non-calculator
1
FFind all the factor pairs of each number.
(a)8
(b)15
(c)20
(d)28
2
FList all the factors of each number, in order.
(a)10
(b)16
(c)18
(d)21
3
FIs the first number a factor of the second number? Write Yes or No.
(a)Is 6 a factor of 24?
(b)Is 5 a factor of 36?
(c)Is 8 a factor of 56?
(d)Is 9 a factor of 72?
(e)Is 7 a factor of 50?
(f)Is 12 a factor of 96?
4
FSquare numbers have a factor pair with the same number twice. Find all the factors of each square number.
(a)36
(b)49
(c)25
5
FMultiply the three numbers. Look for an easy pair to multiply first.
(a)2 × 5 × 7
(b)4 × 3 × 5
(c)2 × 3 × 4
(d)5 × 2 × 9
(e)6 × 3 × 2
(f)2 × 5 × 8
(g)3 × 3 × 4
(h)4 × 5 × 5
6
PSFactor pairs of 24. Mr Ali has 24 chairs. He puts them in equal rows. Write all the ways he could arrange them in rows with the same number in each row.
7
PSA baker makes 3 trays of buns. Each tray has 4 rows with 6 buns in each row. How many buns is that altogether?
8
PSA shop has 4 crates. Each crate has 5 rows of 6 bottles. How many bottles are there altogether?
9
RAmira says, “Every number has exactly two factors: 1 and the number itself.” Is Amira right? Give an example to explain.
10
ROne of these numbers is not a factor of 24. Which one?   1   2   3   4   6   8   9   12
11
RAlways, sometimes or never?
(a)1 is a factor of every number.
(b)A factor of 12 is also a factor of 24.
(c)A factor of a number is smaller than the number.
12
RZoe says, “30 has more factors than 24 because 30 is bigger.” Is Zoe right?
13
RSome numbers have exactly three factors. 9 is one: its factors are 1, 3 and 9. Find all the numbers up to 50 that have exactly three factors. (Hint: they are all square numbers.)
14
RChoose three different numbers from 2, 3, 4 and 5. Multiply them together. Which three numbers make 60?

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