2.2 Introduction and Key Insights

Multiples and factors are related concepts. Instead of saying "three is a factor of 12" we could write "12 is a multiple of three." In general we say that one number $m$ is a multiple of another number $n$ when $m = n \times (another integer)$. We can arrange the (positive) multiples of three, for example, in a list like this one:

$3=3\times 1$
$6=3\times 2$
$9=3\times 3$
$12=3\times 4$

Note that this list should continue on forever, since there are infinitely many multiples of three. You might be familiar with the concept of multiples using a different name, counting by. Counting by threes is simply listing the consecutive multiples of three. Every number will have infinitely many multiples, and sometimes different numbers have overlapping or common multiples. For example, the first four multiples of two are two, four, six, eight. We can see that six is both a multiple of two and a multiple of three.

The least common multiple (LCM) of a collection of numbers is the smallest (or least) multiple shared by every number in the collection. In the above example, the least common multiple of three and two is six. In other words: LCM(3, 2) = 6.
We can also compute the least common multiples in a table, like we did for GCFs, only this time we write out a few of the multiples of our numbers, rather than all of their factors.

For example, let’s explore the LCM(12, 8).

We can extend the entries in the table indefinitely since there are an infinite number of (positive) multiples. However, since we are only interested in the least or smallest one, we can stop entering values in the table as soon as we find a common value.

Our table goes on a bit longer than we need to and shows that 48 is a common multiple of 12 and 8. It also shows a smaller common multiple: 24, which is the least common multiple of 12 and 8. This is the answer to our egg-and-buns question from the beginning of this unit: you need to purchase two dozen eggs and three packages of buns.