Websink Prep UKKS2 MathsYear 6 chaptersMaths textbooksYear 1Year 2Year 4Year 5Year 6GCSEBrowse allPast papersFind a tutorBecome a tutor
Primary Maths · Year 6Chapter 3 · Common factors, multiples and primes
Year 6 · HCF, LCM and Prime Numbers · Lesson 3.6

Year 6: Problems and reasoning with factors, multiples and primes

Part of the Year 6 hcf, lcm and prime numbers chapter. Chapter overview · whole chapter on one page

3.6

Problems and reasoning with factors, multiples and primes

Year 6
Solve problems using common factors, common multiples and the HCF and LCM, Use prime numbers in reasoning, Explain whether a statement is always, sometimes or never true

Many problems are about factors or multiples, but they do not say so. Read the question carefully and decide which one you need.

Use the HCF when things are shared or cut into the largest equal parts, such as the biggest square tile or the most bags. Use the LCM when things happen together again, or when you need the smallest number that fits, such as the fewest packets.

36 cm24 cm
A 36 cm by 24 cm rectangle, covered with square tiles of side 12 cm. 12 is the HCF of 36 and 24, so it is the largest tile that fits exactly.

Reasoning is about whether a statement is always, sometimes or never true. One example that breaks a rule is enough to show that it is not always true. Several examples that work do not prove it is always true. You need a reason.

Key ideas

  • Use the HCF for the largest equal groups, the largest equal pieces, and the biggest square tile.
  • Use the LCM for things that happen together again, and for the smallest number that is a multiple of each number.
  • A prime number of counters can only be arranged in one row or in one column, because its only factors are 1 and itself.
  • A counter-example is one example that breaks the rule. It proves that the statement is not always true.
Worked example 3.6.1
A rectangle is 36 cm long and 24 cm wide. It is covered with the largest possible square tiles, with no gaps. What is the side of each tile, and how many tiles are there?
36 cm24 cm

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

Method. The biggest tile must fit exactly along both sides. Find the HCF of the two lengths, then count the tiles.

  1. Find the largest tile that fits both sides. The tile side must divide exactly into both lengths, so it is the HCF of the two lengths.
    HCF of 36 and 24 = 12 cm
  2. Count the tiles along each side. Divide each length by 12.
    36 ÷ 12 = 3, and 24 ÷ 12 = 2
  3. Multiply to find the total. Tiles along one side times tiles along the other side.
    3 × 2 = 6 tiles

Check. 12 × 3 = 36 and 12 × 2 = 24. ✓

Watch out. Using a smaller tile also fits exactly, but it is not the largest tile that fits.

Worked example 3.6.2
Biscuits come in packets of 8 and in packets of 12. Ali wants the same number of biscuits from each kind, with none left over. What is the fewest biscuits he can have?

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

Method. List the multiples of each number until they meet. The first number in both lists is the LCM.

  1. Write the multiples of 8. Count on in steps of the number. Keep going until you reach a number that is also in the other list.
    8, 16, 24
  2. Write the multiples of 12. Stop at the same number.
    12, 24
  3. Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.
    LCM = 24

Check. 24 ÷ 8 = 3 and 24 ÷ 12 = 2, so 24 is in both times tables. ✓

Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.

Worked example 3.6.3
Find the smallest number that is a multiple of 4, 6 and 9.

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

Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.

  1. Find the LCM of 4 and 6. Use the lists of multiples of the two numbers.
    4: 4, 8, 12
    6: 6, 12
    LCM = 12
  2. Find the LCM of 12 and 9. Use the lists of multiples of the two numbers.
    12: 12, 24, 36
    9: 9, 18, 27, 36
    LCM = 36
  3. Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.
    LCM of 4, 6, 9 = 36

Check. 36 ÷ 4 = 9 and 36 ÷ 6 = 6 and 36 ÷ 9 = 4. ✓

Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.

Watch out

Read the words carefully. “Largest”, “biggest” or “most equal” usually means the HCF. “Smallest”, “fewest” or “next time together” usually means the LCM. Do not just multiply the numbers.

Exercise 3F  Problems and reasoning

Non-calculator
1
PSRuby has 45 red sweets and 60 green sweets. She packs them into bags. Every bag has the same number of red sweets and the same number of green sweets, and no sweets are left over.
(a)What is the largest number of bags she can make?
(b)How many red sweets are in each bag?
(c)How many green sweets are in each bag?
2
PSA rectangle is 60 cm long and 84 cm wide. Square tiles cover it with no gaps. What is the side of the largest tile that fits exactly? Give your answer in cm.
3
PSUsing tiles with the side you found in the last question, how many tiles cover the 60 cm by 84 cm rectangle?
4
PSTwo buses leave a bus station together at 9:00 am. One bus leaves every 12 minutes. The other leaves every 15 minutes. At what time do they next leave together?
5
PSThree lights flash every 4 seconds, every 6 seconds and every 10 seconds. They all flash together at the start. After how many seconds do they next all flash together?
6
RRuby has 47 counters. She wants to arrange all of them in equal rows, with more than one row and more than one counter in each row. Is this possible? Explain your answer.
7
PSWhat is the smallest number that can be divided exactly by 2, 3, 5 and 7?
8
PSI am a common multiple of 9 and 12. I am between 40 and 100. What number am I?
9
PSA cook has 48 apples and 30 pears. She makes identical fruit bowls. Each bowl has the same number of apples and the same number of pears, and all the fruit is used. What is the largest number of bowls she can make?
10
RAlways, sometimes or never?   If two numbers have an HCF of 1, then both numbers are prime.
11
RElla says the LCM of 4 and 10 is 40, because 4 × 10 = 40. Explain Ella’s mistake.
12
RFind the smallest number greater than 100 that is a multiple of 6 and has no prime factors other than 2 and 3.

Websink Prep UK is an independent site. It is not affiliated with, endorsed by or sponsored by AQA, Pearson Edexcel, OCR, WJEC, Eduqas, CCEA, SQA, GL Assessment, CEM or the Department for Education. All exam board names and trademarks belong to their owners. Past papers are linked from the exam boards' own websites and are not hosted here.

About Websink Prep UK · Find a tutor · Privacy · Terms · Cookie settings