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Factors and Multiples

What you'll learn

  • Spot a multiple: a number in a times table, which is a list made by counting in equal jumps.
  • Find a factor: a number that divides another number exactly, with nothing left over.
  • Recognise prime numbers: numbers bigger than 1 with exactly two factors.
  • Choose correct answers from a short list using simple checks.

Multiples: numbers in a times table

Definition

Multiple

A multiple of a number is an answer in that number’s times table. For example, multiples of 5 include 5, 10, 15, 20 and 25.

Multiples of 5 are shown as equal jumps along a number line.

Multiples of 5 are shown as equal jumps along a number line.

To find multiples, count up in equal jumps. For multiples of 6, you count 6, 12, 18, 24, 30, and so on.

Key Idea

Multiples go on

Multiples keep going forever. If you only need one multiple, there may be lots of correct answers.

Example

Find one multiple in a gap

Find one multiple of 6 between 25 and 35.

Counting in 6s shows that 30 is the only multiple of 6 inside the gap from 25 to 35.

Counting in 6s shows that 30 is the only multiple of 6 inside the gap from 25 to 35.

  1. Count in 6s: 6, 12, 18, 24, 30, 36.

  2. Look for a number bigger than 25 and smaller than 35.

  3. The number that fits is 30, so a correct answer is 30.

Tip

Between

If a question says “between 25 and 35”, choose a number inside the gap. Do not use 25 or 35 unless the question clearly allows the end numbers.

Even and odd numbers

Definition

Even and odd

An even number ends in 0, 2, 4, 6 or 8. An odd number ends in 1, 3, 5, 7 or 9.

The last digit sorts whole numbers into even and odd numbers.

The last digit sorts whole numbers into even and odd numbers.

Example

Find the first even multiple

Find the first even multiple of 7.

  1. Count up in 7s: 7, 14, 21, 28.

  2. Check the last digit. 7 is odd, but 14 is even.

  3. The first even multiple of 7 is 14.

Tip

Start counting from the number

For “first multiple” questions, start with the number itself: 7, 14, 21, and so on.

Factors: numbers that divide exactly

Definition

Factor

A factor of a number divides into it exactly, with nothing left over. For example, 3 is a factor of 18 because 18 divided by 3 gives 6 exactly.

18 counters can be split into 3 equal groups of 6, so 3 is a factor of 18.

18 counters can be split into 3 equal groups of 6, so 3 is a factor of 18.

Factors are often found in pairs. For example, 2 and 6 are both factors of 12 because they multiply together to make 12.

Example

List all the factors

Write down all the factors of 12.

A 12-counter rectangle shows factor pairs for 12, including 3 × 4.

A 12-counter rectangle shows factor pairs for 12, including 3 × 4.

  1. Start with 1 and 12.

  2. Check 2: 12 divided by 2 gives 6, so 2 and 6 are factors.

  3. Check 3: 12 divided by 3 gives 4, so 3 and 4 are factors.

  4. Put the factors in order: 1, 2, 3, 4, 6 and 12.

Common Mistake

Mixing up factors and multiples

A factor goes into the number. A multiple is in the number’s times table. For 12, 3 is a factor, but 24 is a multiple.

Prime numbers

Definition

Prime number

A prime number is a number bigger than 1 with exactly two factors: 1 and itself.

A prime number has exactly two factors, shown here for 7.

A prime number has exactly two factors, shown here for 7.

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17 and 19.

Example

Find prime numbers in a range

Find all the prime numbers between 30 and 40.

Crossing out non-primes from 31 to 39 leaves 31 and 37.

  1. Check the numbers from 31 to 39.

  2. Cross out the even numbers: 32, 34, 36 and 38.

  3. Cross out 33 and 39 because they can be divided by 3 exactly. Cross out 35 because it can be divided by 5 exactly.

  4. The numbers left are 31 and 37, so the prime numbers are 31 and 37.

Common Mistake

1 is not prime

1 has only one factor: itself. A prime number must have exactly two factors, so 1 is not prime.

Tip

The only even prime

2 is the only even prime number. Every other even number has 2 as a factor, so it cannot be prime.

Square numbers

Definition

Square number

A square number is made by multiplying a number by itself. Small square numbers include 1, 4, 9, 16, 25, 36 and 49.

Square numbers can be made as equal rows and columns, such as 5 × 5 = 25.

Example

Choose a square number from a list

From the numbers 18, 20, 22, 25, 28 and 30, choose the square number.

  1. Remember the small square numbers: 1, 4, 9, 16, 25 and 36.

  2. Compare these with the numbers in the list.

  3. The square number in the list is 25.

Choosing answers from a list

Sometimes you are given a list of numbers and asked to choose answers from it. Read carefully: if it says “from the list”, your answer must be in the list.

Key Idea

Use the given list

Do not invent a new number if a list is given. Even if another number would work, your answer must come from the list.

Example

Use a list of numbers

From the numbers 4, 11, 12, 15, 18, 25 and 29, choose:

  1. For a factor of 20, choose 4 because 20 divided by 4 gives 5 exactly.

  2. For a multiple of 6, 12 or 18 would work, so one correct answer is 12.

  3. For all the prime numbers, choose 11 and 29.

Combining clues

Some questions give more than one clue. The answer must match every clue, not just one of them.

Example

Use all the clues

A number is odd. It is a factor of 30 and a multiple of 5. Find the two possible numbers.

Filtering the factors of 30 by odd numbers and then multiples of 5 leaves 5 and 15.

Filtering the factors of 30 by odd numbers and then multiples of 5 leaves 5 and 15.

  1. List the factors of 30: 1, 2, 3, 5, 6, 10, 15 and 30.

  2. Keep only the odd ones: 1, 3, 5 and 15.

  3. From those, keep the multiples of 5: 5 and 15.

  4. The two possible numbers are 5 and 15.

Adding prime numbers

Definition

Sum

The sum of two numbers is the answer you get when you add them.

Tip

Odd totals

For an odd total made from two prime numbers, try 2 first because 2 is the only even prime number.

Example

Two primes with a given sum

Find two prime numbers with a sum of 21.

  1. 21 is odd, so try using 2 as one of the prime numbers.

  2. Take 2 away from 21. The other number is 19.

  3. 19 is prime.

  4. So the two prime numbers are 2 and 19.

Exam technique

In the exam

  1. Underline key words such as factor, multiple, prime, even, odd and between.

  2. If a list of numbers is given, make sure every answer comes from that list.

  3. Check how many answers are needed: one, two, or all of them.

Self review

Check yourself

  • Can you list three multiples of 8?

  • Can you write all the factors of 16?

  • Can you explain why 21 is not a prime number?

Recap questions

1 of 5

Which option is a multiple of 8 between 30 and 45?

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Factor pairs pointing into 12 and multiples of 12 extending outward on a number line A factor goes into a number exactly, with no remainder. A multiple is a number in that number's times table.

For 12, the factors are 1, 2, 3, 4, 6 and 12 because each divides 12 exactly. The multiples start 12, 24, 36 and 48 because they come from 12×112 \times 112×1, 12×212 \times 212×2, 12×312 \times 312×3 and 12×412 \times 412×4.

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A multiple of a number is an answer in that number’s [     ].

Factors and Multiples Revision Guide

  1. GCSE
  2. /Maths
  3. /Factors and Multiples