The terms factor and multiple are sometimes confused with each other. There are infinitely many multiples of 10, including 30, 70, 20, and also 0, -40, and so on. The number 10 has only four factors: 2 and 5, and the "trivial" factors 1 and 10. See more below and at factor.
A multiple of a number is any product of that number and an integer. For example, the multiples of 5 include 5 x 0, 5 x 1, 5 x 2, 5 x 3, and so on. (And, when children are able to use negative numbers, multiples will include 5 x (-1), 5 x (-2), and so on.
- Keeping the concept clear: When naming the multiples of a number, children (and adults!) often forget to include the number, itself, and are often unsure whether to include 0 or not. The multiples of 3 include 3 times any integer, including 3 x 0 and 3 x 1. So 3 "is a multiple of 3" (though a pretty trivial one) and 5 "is a multiple of 5" (again, trivial). Zero is a multiple of every number, though, again, a multiple that's not terribly informative. In particular, 0 is an even number, because it is a multiple of 2. But, because 0 is a multiple of anything, it is often not useful to list. And when we are asking for the "smallest" multiple (for example, the "least common multiple"), we include only positive multiples.
- Keeping the language clear: It's imprecise to refer to a number as "a multiple" without saying what it is a multiple of. The number 12 is a "multiple of 4" or a "multiple of 6" but not just "a multiple." (It is not, for example, "a multiple" of 5.) Numbers are multiples of something, not just "multiples."
- A fine point: The term multiple -- like factor and divisible -- is generally used only to refer to results of multiplication by a whole number.
It is often useful to know what multiples two numbers have in common. One way is to list (some of) the multiples of each and look for a pattern. For example, to find the common (positive) multiples of 4 and 6, we might list:
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, ...
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...
The numbers 12, 24, 36, and 48 appear on both of these lists (and more would appear if the lists were longer). They are common multiples. The least common multiple is the smallest of these: 12. All the other common multiples are, themselves, multiples of the least common multiple.
Another way of finding the least common multiple of 4 and 6 involves factoring both numbers into their prime factors. The prime factorization of 4 is 2 x 2, and the prime factorization of 6 is 2 x 3. Any common multiple of 4 and 6 will need enough prime factors to make each of these numbers. So, it will need two 2s and a 3 -- the two 2s to make 4 (as 2 x 2) and the 3 (along with one of the 2s) to make 6 (as 2 x 3). The prime factorization of this least common multiple is, therefore, 2 x 2 x 3, and the least common multiple is 12.
Related mathematical terms
See factor, product, and divisor for more.
What's in a word?
A multiple is what you get by multiplying.