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Academy11+

11+ Maths · Topic explained

Factors, multiples and primes

Why can 12 sweets be shared out fairly in lots of ways, but 7 can't?

The big idea

Factors divide into a number exactly; multiples are in its times table; a prime has exactly two factors.

Picture it

Arrange 12 sweets in neat rectangles: 1 × 12, 2 × 6, 3 × 4. The sides are factors. 7 only makes a line.

Key words

factor
A number that divides into another exactly: 3 is a factor of 12.
multiple
A number in a times table: 12, 18 and 24 are multiples of 6.
prime number
A number with exactly two factors, 1 and itself: 2, 3, 5, 7, 11…
common
Shared by both numbers, as in a common factor or common multiple.

Worked examples

  1. 1Warm-up

    List the factors of 12

    • Find them in pairs: 1 × 12, 2 × 6, 3 × 4
    • Factors of 12: 1, 2, 3, 4, 6, 12
  2. 211+ standard

    Which is prime: 21, 27, 29 or 33?

    • 21 = 3 × 7, 27 = 3 × 9, 33 = 3 × 11
    • 29 has no factors except 1 and 29
    • 29 is prime
  3. 311+ standard

    Find the lowest common multiple of 6 and 8

    • Multiples of 6: 6, 12, 18, 24…
    • Multiples of 8: 8, 16, 24…
    • Lowest common multiple: 24
  4. 411+ standard

    Find the highest common factor of 18 and 30

    • Factors of 18: 1, 2, 3, 6, 9, 18
    • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
    • Highest common factor: 6

The common mistake

The mistake

9 and 21 are prime, because they're odd.

Instead

9 = 3 × 3 and 21 = 3 × 7, so neither is prime. And 2 is prime, even though it's even.

Why: Prime means exactly two factors, nothing to do with odd or even. (1 is not prime: it has only one factor.)

Where it comes up in the 11+

Listing factors, spotting primes, and common multiples in puzzles like 'when do two buses next leave together?'

The video

Video coming soon

We're making a short video that explains factors, multiples and primes step by step. Until then, everything it covers is on this page.

Free printable

Factors, multiples and primes worksheet

Practice questions on this topic to print and try on paper, at home or on the way to school.

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Quick quiz

Have a go first, then tap to see the answer.

  1. 1. How many factors does 16 have?

    • 4
    • 5
    • 6
    • 8
    Show the answer

    5

    1, 2, 4, 8 and 16 (4 × 4 counts once).

  2. 2. Which are the prime numbers between 20 and 30?

    • 21 and 23
    • 23 and 29
    • 23, 27 and 29
    • 25 and 29
    Show the answer

    23 and 29

    21 = 3 × 7, 25 = 5 × 5, 27 = 3 × 9; the even ones divide by 2.

  3. 3. Buses leave every 10 minutes and every 15 minutes. Both leave at 9:00. When do they next leave together?

    • 9:15
    • 9:25
    • 9:30
    • 10:00
    Show the answer

    9:30

    The lowest common multiple of 10 and 15 is 30, so 30 minutes later.