Multiples

A number m is said to be a multiple of another number n if dividing m by n leaves no remainder, that is, if the division is exact. $$ \frac{m}{n} = q \ \text{with remainder } 0 $$

For every natural number, there are infinitely many multiples.

They are obtained by multiplying the given number by k, where k is any natural number.

$$ m = n \cdot k \ \ \forall k = 1,2,3,\ldots $$

The only exception is the number zero, which has itself as its only multiple. This is because zero is an absorbing element under multiplication: any number k multiplied by zero is always equal to zero.

A practical example

The number 6 is a multiple of 3 because dividing 6 by 3 yields a remainder of 0.

$$ \frac{6}{3} = 2 \ \text{with remainder } 0 $$

The number 3 has infinitely many multiples for k = 1, 2, …

$$ 3 \ , \ 6 \ , \ 9 \ , \ 12 \ , \ 15 \ , \ \ldots $$

Note. An odd number alternates between even and odd multiples. An even number, by contrast, has only even multiples, because the product of an odd number and an even number is always even. For example, the multiples of the number 2 are $$ 2 \ , \ 4 \ , \ 6 \ , \ 8 \ , \ 10 \ , \ \ldots $$

And so on.