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Exam questions · Maths · Number Without a Calculator

Factors, Multiples and Primes

  • 7 exam questions
  • 18 marks
  • 9 quick checks
  1. 1 Write down [1 mark]

    Write down a multiple of 7 that is between 50 and 60.

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    Model answer

    \(7 \times 8 = 56\), so 56 is the only multiple of 7 between 50 and 60.

    Mark scheme

    • 56 — B1
  2. 2 Write down [2 marks]

    Write down all the prime numbers between 40 and 50.

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    Model answer

    \(41, 43, 47\). The other numbers are not prime: \(42\), \(44\), \(46\), \(48\) are even, \(45\) is a multiple of 5 and \(49 = 7 \times 7\).

    Mark scheme

    • Two correct primes and no more than one incorrect number — B1
    • 41, 43 and 47 only — B1
  3. 3 Write [3 marks]

    Write 252 as a product of its prime factors. Give your answer in index form.

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    Model answer

    \(252 = 4 \times 63 = 2 \times 2 \times 7 \times 9 = 2 \times 2 \times 3 \times 3 \times 7 = 2^2 \times 3^2 \times 7\).

    Mark scheme

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

    (a) Find the highest common factor (HCF) of 36 and 60. [2 marks] (b) Find the lowest common multiple (LCM) of 36 and 60. [2 marks]

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    Model answer

    \(36 = 2^2 \times 3^2\) and \(60 = 2^2 \times 3 \times 5\). (a) HCF \(= 2^2 \times 3 = 12\). (b) LCM \(= 2^2 \times 3^2 \times 5 = 180\).

    Mark scheme

    • (a) Correct list of factors, or the prime factors of both numbers — M1
    • (a) 12 — A1
    • (b) Correct list of multiples, or a product that uses every prime once at its highest power — M1
    • (b) 180 — A1
  5. 5 Work out [2 marks]

    Is 91 a prime number? You must show your working.

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    Model answer

    No. \(91 = 7 \times 13\), so 91 has factors other than 1 and itself.

    Mark scheme

    • Finds a factor of 91 other than 1 and 91, such as \(91 \div 7 = 13\) — M1
    • No, with \(7 \times 13\) — Q1
  6. 6 Work out [3 marks]

    Lighthouse A flashes every 15 seconds. Lighthouse B flashes every 20 seconds. They both flash at 8 pm. At what time do they next flash together?

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    Model answer

    The LCM of 15 and 20 is 60. Multiples of 15: 15, 30, 45, 60. Multiples of 20: 20, 40, 60. They next flash together after 60 seconds, which is at 8.01 pm.

    Mark scheme

    • A list of multiples of each, or a prime factor method — M1
    • 60 seconds — A1
    • 8.01 pm — B1
  7. 7 Work out [3 marks]

    The highest common factor (HCF) of two numbers is 12. Their lowest common multiple (LCM) is 180. One of the numbers is 36. Work out the other number.

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    Model answer

    For any two numbers, the product of the numbers equals the HCF multiplied by the LCM. So \(36 \times n = 12 \times 180 = 2160\), and \(n = 2160 \div 36 = 60\).

    Mark scheme

    • \(12 \times 180 = 2160\) — M1
    • \(2160 \div 36\) — M1
    • 60 — 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.