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Exam questions · Maths · Number Without a Calculator

Factors, Multiples and Primes

  • 6 exam questions
  • 18 marks
  • 9 quick checks
  1. 1 Write down [2 marks]

    Here is a list of numbers. \(6\) \(9\) \(16\) \(17\) \(27\) \(30\) (a) Write down a cube number from the list. [1 mark] (b) Write down a prime number from the list. [1 mark]

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    Model answer

    (a) \(27 = 3^3\). (b) \(17\) is prime, because its only factors are 1 and 17.

    Mark scheme

    • (a) 27 — B1
    • (b) 17 — B1
  2. 2 Write [3 marks]

    Write 126 as a product of prime factors. Give your answer in index form.

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    Model answer

    \(126 = 2 \times 63 = 2 \times 7 \times 9 = 2 \times 3 \times 3 \times 7 = 2 \times 3^2 \times 7\).

    Mark scheme

    • A correct first step, such as a factor tree with a correct first pair — M1
    • \(2 \times 3 \times 3 \times 7\) — A1
    • \(2 \times 3^2 \times 7\) — A1
  3. 3 Work out [4 marks]

    (a) Find the highest common factor (HCF) of 56 and 84. [2 marks] (b) Find the lowest common multiple (LCM) of 56 and 84. [2 marks]

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    Model answer

    \(56 = 2^3 \times 7\) and \(84 = 2^2 \times 3 \times 7\). (a) HCF \(= 2^2 \times 7 = 28\). (b) LCM \(= 2^3 \times 3 \times 7 = 168\).

    Mark scheme

    • (a) A correct list of factors or prime factors of both numbers — M1
    • (a) 28 — A1
    • (b) A correct list of multiples or a correct product of primes — M1
    • (b) 168 — A1
  4. 4 Explain [2 marks]

    (a) Aisha says, ‘All prime numbers are odd.’ Explain why Aisha is wrong. [1 mark] (b) Sam says, ‘The number 51 is prime.’ Show that Sam is wrong. [1 mark]

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    Model answer

    (a) 2 is a prime number and it is even. (b) \(51 = 3 \times 17\), so 51 has factors other than 1 and itself. The digits add to 6, so 51 is divisible by 3.

    Mark scheme

    • (a) Identifies that 2 is an even prime — B1
    • (b) \(3 \times 17\), or shows that 51 is divisible by 3 — B1
  5. 5 Work out [3 marks]

    A cog with 24 teeth meshes with a cog with 40 teeth. A tooth on each cog is marked, and at the start the two marked teeth are touching. The cogs turn. How many complete turns does each cog make before the two marked teeth next touch?

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    Model answer

    The marked teeth next touch after a number of teeth that is a common multiple of 24 and 40. \(24 = 2^3 \times 3\) and \(40 = 2^3 \times 5\), so the LCM is \(2^3 \times 3 \times 5 = 120\). The smaller cog turns \(120 \div 24 = 5\) times and the larger cog turns \(120 \div 40 = 3\) times.

    Mark scheme

    • Identifies that 120 teeth have passed, or a correct list of multiples of 24 and 40 — M1
    • 120 — A1
    • 5 turns of the smaller cog and 3 turns of the larger cog — B1
  6. 6 Work out [4 marks]

    \(N = 2^3 \times 3 \times 5^2\) (a) Work out the value of \(N\). [2 marks] (b) Is \(N\) a multiple of 45? You must give a reason for your answer. [2 marks]

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    Model answer

    (a) \(2^3 = 8\) and \(5^2 = 25\), so \(N = 8 \times 3 \times 25 = 600\). (b) No. \(45 = 3^2 \times 5\), which needs two factors of 3, but \(N\) only has one. Also \(600 \div 45 = 13.33\ldots\), which is not a whole number.

    Mark scheme

    • (a) \(8 \times 3 \times 25\) or two of \(8, 3, 25\) multiplied — M1
    • (a) 600 — A1
    • (b) Writes \(45 = 3^2 \times 5\) or attempts \(600 \div 45\) — M1
    • (b) No, with a correct reason — A1

Quick check

  1. 1

    What is the smallest whole number that 72 must be multiplied by to give a square number?

    1. A2
    2. B9
    3. C6
    4. D3
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    A: 2

    \(72 = 2^3 \times 3^2\). The index of 2 is odd, so multiply by 2 to get \(144 = 12^2\).

  2. 2

    A teacher has 48 red pens and 72 blue pens. What is the greatest number of identical packs that she can make using all of the pens?

    1. A6
    2. B12
    3. C144
    4. D24
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    D: 24

    The answer is the HCF of 48 and 72, which is 24.

  3. 3

    Which of these numbers is prime?

    1. A63
    2. B51
    3. C59
    4. D57
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    C: 59

    59 has no factors other than 1 and 59. 51 = 3 × 17, 57 = 3 × 19 and 63 = 7 × 9.

  4. 4

    What is the highest common factor (HCF) of 18 and 30?

    1. A6
    2. B9
    3. C3
    4. D90
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    A: 6

    The common factors are 1, 2, 3 and 6, and the highest of these is 6.

  5. 5

    What is the lowest common multiple (LCM) of 6 and 8?

    1. A14
    2. B48
    3. C24
    4. D2
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    C: 24

    Multiples of 8 are 8, 16, 24 and 24 is the first one that is also a multiple of 6.

  6. 6

    Which of these is 72 written as a product of its prime factors?

    1. A\(2 \times 36\)
    2. B\(2^2 \times 3^3\)
    3. C\(2^3 \times 3^2\)
    4. D\(8 \times 9\)
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    C: \(2^3 \times 3^2\)

    72 = 8 × 9 = 2 × 2 × 2 × 3 × 3 = \(2^3 \times 3^2\).

  7. 7

    How many factors does 20 have?

    1. A8
    2. B5
    3. C6
    4. D4
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    C: 6

    The factors are 1, 2, 4, 5, 10 and 20, which makes six.

  8. 8

    Which of these numbers is divisible by 9?

    1. A4520
    2. B4531
    3. C4527
    4. D4528
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    C: 4527

    The digits of 4527 add up to 18, which is a multiple of 9.

  9. 9

    Two lights flash every 8 seconds and every 12 seconds. They flash together now. After how many seconds will they next flash together?

    1. A20
    2. B96
    3. C4
    4. D24
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    D: 24

    The LCM of 8 and 12 is 24, so they next flash together after 24 seconds.