A multiple of a number is the product of that number and an integer. The number itself does not need to be an integer, it can be any number, but in order for the result to be considered a multiple of the number, it needs to be multiplied by an integer; 4 is a multiple of 2 because 2 × 2 = 4, but 4 is not a multiple of 3 because 3 cannot be multiplied by an integer to get a result of 4 (3 × 1.33 = 4).
|Multiples of 4 include 0, 4, 8, 12,...|
|Multiples of 6 include 0, 6, 12, 18,...|
Properties of multiples
- 0 is a multiple of everything, since 0 multiplied by any number is still 0.
- Every integer is a multiple of itself because anything multiplied by 1 is the same number, so any integer can be multiplied by 1 to result in said integer, making it a multiple of itself.
- The product of an integer, n, and any integer, is a multiple of n. This should make sense because it fits the definition of what a multiple is, described above.
- If two integers, designated A and B, are a multiple of another number, n, then A + B and A - B are also multiples of n.
- Example: 4 and 8 are multiples of 2. 8 + 4 = 12; 8 - 4 = 4. Both 4 and 12 are also multiples of 2.
Multiples vs factors
Since both multiples and factors involve multiplication, they are sometimes confused. However, they are not the same thing.
A multiple is the product of a number and an integer. The number does not have to be an integer. For example, 3 is a multiple of 1.5, since 1.5 × 2 is 3. It also works with irrational numbers such as π. 2π is a multiple of π, and so is any other integer multiple of π.
In contrast, a factor, is a whole number by which a larger whole number can be divided evenly. All whole numbers are factors of themselves, since they can be divided by themselves to result in 1.
Both terms in the following multiplication problem are factors of 12:
3 × 4 = 12
A given whole number has many different factors. The following are other factors of 12:
6 × 2 = 12
1 × 12 = 12
1, 2, 3, 4, 6, and 12 are all factors of 12. 12 is also a multiple of any of these numbers, since they can each be multiplied by an integer to result in 12.