Factors and multiples, odd and even numbers

What is a multiple?

A multiple of a number is what you get by multiplying it by a whole number (0, 1, 2, 3…). The multiples of 4, for example, are all the results in the 4 times table: 0, 4, 8, 12, 16, 20… — and this list never ends, there's an infinite number of multiples for every number.

What is a factor?

A factor of a number is a whole number that divides it exactly, with no remainder left over. 3 is a factor of 12 because 12 ÷ 3 = 4 exactly (remainder 0). On the other hand, 5 is NOT a factor of 12, because 12 ÷ 5 = 2 remainder 2.

Unlike multiples, the factors of a number are limited in number. 12 has exactly six factors: 1, 2, 3, 4, 6 and 12.

The link between multiples and factors

Multiplying and dividing are inverse operations — so multiples and factors are inverse ideas too. Saying "3 is a factor of 12" is exactly the same as saying "12 is a multiple of 3". It's the same relationship, seen from both sides.

1. 12 ÷ 3 = 4, with no remainder → 3 divides 12 → 3 is a factor of 12.
2. 3 × 4 = 12 → multiplying 3 gets you back to 12 → 12 is a multiple of 3 (and of 4).

It's this link that makes factors and multiples so useful for simplifying a fraction: finding a common factor of the numerator and denominator lets you reduce the fraction without changing its value.

Odd and even numbers

A number is even if it's a multiple of 2, meaning it can be shared into two perfectly equal groups. Otherwise, it's odd — there's always 1 left over when you try to share it in two.

You don't need to actually divide to find out: just look at the last digit (the units digit) of the number.

• Last digit 0, 2, 4, 6 or 8 → the number is even (e.g. 348, 1,250).
• Last digit 1, 3, 5, 7 or 9 → the number is odd (e.g. 347, 1,251).

Finding all the factors of a number

To avoid missing any, you test whole numbers one by one starting from 1, and each time you note the pair of factors that go together (the number you tested and the result of the division).

Example: all the factors of 24.

1. 24 ÷ 1 = 24 → 1 and 24 are factors.
2. 24 ÷ 2 = 12 → 2 and 12 are factors.
3. 24 ÷ 3 = 8 → 3 and 8 are factors.
4. 24 ÷ 4 = 6 → 4 and 6 are factors.
5. 24 ÷ 5: remainder 4, not a factor — you can stop here, since you've already gone past 4 (the square root of 24), so everything else has already been found.

Tricks for spotting a multiple

• Multiple of 2: the number is even (last digit 0, 2, 4, 6, 8).
• Multiple of 5: the number ends in 0 or 5.
• Multiple of 10: the number ends in 0.
• Multiple of 3: the sum of its digits is a multiple of 3 (e.g. 141 → 1+4+1=6, a multiple of 3 → so is 141).

Mistakes to avoid

• Confusing multiple and factor: a multiple is ALWAYS bigger than or equal to the starting number, a factor is ALWAYS smaller than or equal to it.
• Forgetting 1 and the number itself in the list of factors.
• Stopping too early when looking for factors, forgetting to test up to the square root of the number.
• Looking at the whole number instead of just the last digit to decide whether it's odd or even.

Worked exercises