Multiples and divisors of a number

What is a multiple?

A multiple of an integer n is obtained by multiplying n by another integer. For example, the multiples of 3 are: 0, 3, 6, 9, 12, 15, 18… These are found by calculating 3 × 0, 3 × 1, 3 × 2, 3 × 3, and so on. A number has an infinite number of multiples.

What is a divisor?

A divisor of a number n is an integer that divides n exactly, leaving no remainder. For example, 4 is a divisor of 12 because 12 / 4 = 3 (the result is a whole number; there is no remainder). We also say that 12 is divisible by 4.

The link between multiples and divisors

These two concepts are linked: if a is a multiple of b, then b is a divisor of a. Example: 20 is a multiple of 5 (because 5 × 4 = 20), so 5 is a divisor of 20 (because 20 ÷ 5 = 4).

Finding all the divisors of a number

To list the divisors of 12, we test each integer starting from 1:

Divisor tested	12 / divisor	Divisor of 12?
1	12	yes
2	6	yes
3	4	yes
4	3	yes
5	2.4	no
6	2	yes

The divisors of 12 are therefore: 1, 2, 3, 4, 6 and 12.

Common pitfalls

• 0 is a multiple of every number (n × 0 = 0), but 0 is not a divisor of any number because it is impossible to divide by 0.
• 1 is a divisor of all integers.
• Every integer is both a divisor and a multiple of itself (n / n = 1 and n × 1 = n).