GCSE Mathematics
› M1.2 Factors, multiples, primes, HCF, LCM
Factors, multiples, primes, HCF, LCM
Free GCSE Mathematics practice questions on Factors, multiples, primes, HCF, LCM.
Aligned with the UK Department for Education GCSE subject content — works for any UK GCSE exam board.
Sample questions below with detailed mark schemes. Sign up to practise the full set with spaced repetition.
Calculate — 2 marks
A teacher has 24 red pens and 36 blue pens. She wants to make identical packs for her students, with each pack containing the same number of red pens and the same number of blue pens. She wants to use all the pens with none left over.
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(a) Calculate the greatest number of identical packs the teacher can make.
[1 mark]
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(b) Calculate how many red pens and how many blue pens will be in each pack.
[1 mark]
Show mark scheme
- (a) 12
- (b) 2 red pens and 3 blue pens
Explain — 2 marks
A librarian is organising 24 fiction books and 36 non-fiction books onto shelves. Each shelf must contain the same number of books, and all books on a shelf must be of the same type. The librarian wants to use the largest possible number of books per shelf.
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(a) Work out the maximum number of books that can be placed on each shelf.
[1 mark]
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(b) Explain why 5 books per shelf is not possible.
[1 mark]
Show mark scheme
- (a) 12
- (b) 5 is not a factor of 24 or 5 is not a factor of 36 (or equivalent)
Describe — 3 marks
A youth club has 36 members and wants to split them into equal-sized teams for a competition. Each team must have more than 4 members but fewer than 12 members.
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(a) List all the possible team sizes that the youth club could use.
[2 marks]
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(b) Describe why a team size of 5 would not work.
[1 mark]
Show mark scheme
- (a) Lists factors of 36 that are greater than 4 and less than 12: 6, 9 (one mark for each correct value, or both marks for complete list)
- (b) States that 5 is not a factor of 36 (or 36 is not divisible by 5 / 36 ÷ 5 = 7.2 which is not a whole number)
State — 4 marks
A bathroom wall measures 48 cm by 72 cm. A tiler wants to cover the wall using identical square tiles with whole number side lengths.
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(a) State all the common factors of 48 and 72.
[2 marks]
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(b) State the largest possible side length of a square tile that can be used to cover the wall exactly with no gaps or overlaps.
[1 mark]
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(c) The tiler decides instead to use rectangular tiles measuring 8 cm by 12 cm. State the smallest number of these tiles needed to form a square arrangement.
[1 mark]
Show mark scheme
- (a) Lists common factors: 1, 2, 3, 4, 6, 8, 12, 24
- (b) 24 (cm)
- (c) 6
Compare — 3 marks
A teacher has 48 red counters and 60 blue counters. They want to divide the counters into identical sets with no counters left over, using the largest possible set size. Each set must contain only one colour.
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(a) Find the highest common factor (HCF) of 48 and 60.
[1 mark]
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(b) Compare the number of sets that can be made from the red counters with the number of sets that can be made from the blue counters.
[2 marks]
Show mark scheme
- (a) HCF = 12
- (b) Correct number of sets: 4 (red) and 5 (blue)
- (b) Correct comparison (e.g., 5 > 4, or blue makes 1 more set, or equivalent)
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