OpenRevise

Exam questions · Maths · Number Without a Calculator

Factors, Multiples and Primes

  • 6 exam questions
  • 19 marks
  • 9 quick checks
  1. 1 List [2 marks]

    List all the factors of 36.

    Show answerHide answer

    Model answer

    \(1, 2, 3, 4, 6, 9, 12, 18, 36\). They come in pairs: \(1 \times 36\), \(2 \times 18\), \(3 \times 12\), \(4 \times 9\) and \(6 \times 6\).

    Mark scheme

    • At least 5 correct factors with no more than one incorrect extra — M1
    • All nine factors and no others — A1
  2. 2 Write down [2 marks]

    Here is a list of numbers. \(15\) \(21\) \(23\) \(27\) \(35\) \(41\) \(51\) From the list, write down all the prime numbers.

    Show answerHide answer

    Model answer

    \(23\) and \(41\). \(51 = 3 \times 17\), so it is not prime, and the other numbers all have a factor other than 1 and themselves.

    Mark scheme

    • Either 23 or 41 with no more than one incorrect extra — B1
    • Both 23 and 41 and no others — B1
  3. 3 Work out [3 marks]

    Here is part of a factor tree for 84. The circled letters A and B stand for missing numbers. (a) Work out the value of A and the value of B. (2 marks) (b) Write 84 as a product of its prime factors. Give your answer in index form. (1 mark)

    A factor tree for 84 with two missing values, A (the other factor of 84) and B (the other factor of 4).
    Show answerHide answer

    Model answer

    (a) \(A = 84 \div 4 = 21\) and \(B = 4 \div 2 = 2\). (b) \(84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7\).

    Mark scheme

    • A = 21 — B1
    • B = 2 — B1
    • \(2^2 \times 3 \times 7\) — B1
  4. 4 Express [3 marks]

    Express 540 as a product of its prime factors. Give your answer in index form.

    Show answerHide answer

    Model answer

    \(540 = 54 \times 10 = 2 \times 27 \times 2 \times 5\), so \(540 = 2^2 \times 3^3 \times 5\).

    Mark scheme

    • Starts a correct factor tree or repeated division, with at least two correct stages — M1
    • \(2 \times 2 \times 3 \times 3 \times 3 \times 5\), or a product of primes with at most one error — M1
    • \(2^2 \times 3^3 \times 5\) — A1
  5. 5 Find [6 marks]

    (a) Express 48 as a product of its prime factors. (2 marks) Given that \(84 = 2^2 \times 3 \times 7\), (b) find the highest common factor (HCF) of 48 and 84. (2 marks) (c) find the lowest common multiple (LCM) of 48 and 84. (2 marks)

    Show answerHide answer

    Model answer

    (a) \(48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3\). (b) The common primes are \(2^2\) and 3, so the HCF is \(4 \times 3 = 12\). (c) Take the highest power of every prime: \(2^4 \times 3 \times 7 = 16 \times 21 = 336\).

    Mark scheme

    • (a) Correct method to factorise 48 — M1
    • (a) \(2^4 \times 3\) — A1
    • (b) Identifies the common factors \(2^2\) and 3, or \(2 \times 2 \times 3\) — M1
    • (b) 12 — A1
    • (c) \(2^4 \times 3 \times 7\) or a correct list of multiples — M1
    • (c) 336 — A1
  6. 6 Work out [3 marks]

    Tom visits his grandmother every 6 days. Lisa visits her every 8 days. They both visited on 1 June. On what date will they next both visit their grandmother?

    Show answerHide answer

    Model answer

    The days they both visit are common multiples of 6 and 8. The multiples of 6 are 6, 12, 18, 24 and those of 8 are 8, 16, 24, so the LCM is 24. 24 days after 1 June is 25 June.

    Mark scheme

    • A list of at least three multiples of each number, or the prime factors of both — M1
    • 24 — A1
    • 25 June — C1

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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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\(8 \times 9\)
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    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
    Show answerHide answer

    D: 24

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