Topics: Mathematical Language

Mathematical Domain: Number & Operations in Base Ten, Ratios & Proportional Relationships

Question: Children (and adults) are often uncertain whether the multiples of, say, 12 are the numbers one can multiply (like 3 and 4) to make 12, or the numbers that one can make by multiplying 12 times other numbers. The terms *multiple* and *factor* are often confused. What *are* the multiples of a number?

By example:

Multiples of 3, like …–9, –6, –3, 0, 3, 6, 9, 12, 15… are formed by multiplying 3 by any integer (a “whole” number, negative, zero, or positive, such as…–3, –2, –1, 0, 1, 2, 3…).

Multiples of 12, like …–36, –24, –12, 0, 12, 24, 36, 48, 60…, are all 12 ×

n, wherenis an integer.Multiples of 2, like …–8, –6, –4, –2, 0, 2, 4, 6, 8, 10, 12…, are all even, 2 × any integer.

Generally:

The multiples of an integer are all the numbers that can be made by multiplying that integer by any integer. Because 21 can be written as 3 × 7, it is a multiple of 3 (and a multiple of 7).

Though 21 can also be written as 2 × 10, it is not generally considered a multiple of 2 (or 10), because the word multiple is generally (

alwaysin K–12 mathematics) used only in the context of integers.

**Keeping the**When naming the multiples of a number, children (and adults!) often forget to include the number, itself, and are often unsure whether or not to include 0. The multiples of 3 include 3 times*concept*clear:*any*integer, including 3 × 0 and 3 × 1. So 3 “is a multiple of 3” (though a trivial one) and 5 “is a multiple of 5” (again, trivial). Zero is a multiple of*every*number so (among other things) it is an even number. When asked for the “smallest” multiple (for example, the*least common multiple*), the implication is that only*positive*multiples are meant. Thus 6 is the “least” common multiple of 3 and 2 even though 0 and –6 (and so on) are also multiples that 3 and 2 have in common, and they are less than 6.**Keeping the**It is imprecise to refer to a number as “a multiple” without saying what it is a multiple*language*clear:*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.”

Also, 6 is a*factor*of 12, not a multiple of 12. And 12 is a multiple of 6, not a factor of 6.**A fine point:**The term—like*multiple**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*, multiples that the two numbers have in common. The *least common multiple* is the smallest of these: 12. All the other common multiples are 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 × 2, and the prime factorization of 6 is 2 × 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 one 3—the two 2s that are needed to make 4 (as 2 × 2) and the 3 (along with one of the 2s we already have) to make 6 (as 2 × 3). The prime factorization of this least common multiple is, therefore, 2 × 2 × 3, and the least common multiple is 12.

A *multiple* is what you get by *multiplying*.