Welcome to lesson three of Foundations of Factors and Multiples — only one more to go after this! In the first two lessons, you built a solid toolkit for taking a number apart: you learned how to verify factor relationships, find factor pairs, and assemble complete factor lists. All of that work answered one central question: What divides evenly into this number?

Now we flip the perspective entirely. Instead of looking inward at what divides a number, we look outward at what a number produces when we multiply. That shift in viewpoint leads us to the idea of multiples and reveals a powerful connection that ties together everything you have learned so far.

So far our focus has been on looking inside a number. Given $36$, for example, we searched for smaller numbers that divide it exactly, and we called those its factors. But every time we wrote a multiplication fact like $4 \times 9 = 36$, we were also saying something about $36$ from the point of view of $4$: the number $36$ is a result of multiplying by a whole number.

A multiple of a whole number is the result of multiplying that number by any whole number ($0, 1, 2, 3, \ldots$).

While zero is technically a multiple of every number (because any number times $0$ equals $0$), it isn't very helpful when we are trying to plan a schedule or share items. For this reason, in this course, we will focus on positive multiples — the results of multiplying by $1, 2, 3, \dots$ and so on.

To list multiples, simply start at the number and keep adding it. Let's generate the first six multiples of $8$:

Sometimes we do not need to generate a whole list. Instead, we just need to answer a yes-or-no question: Is $56$ a multiple of $7$? The test is straightforward — divide and check the remainder:

Because the division comes out exactly, $56$ a multiple of . Compare that with remainder . The remainder is not zero, so is a multiple of .

Factors and multiples describe the same relationship viewed from opposite sides:

• If $A$ is a factor of $B$, then $B$ is a multiple of $A$.
• If $B$ is a multiple of $A$, then is a of .

Imagine a bakery that packages muffins in boxes of $6$. The batch sizes that fill boxes perfectly are $6, 12, 18, 24, 30, \ldots$ — exactly the multiples of $6$. If a customer orders $42$ muffins, we can check: with no remainder, so is a multiple of and the order fills complete boxes with none left over.

In this lesson, you learned that a multiple of a number is the product of that number and any positive whole number, and that the list of multiples stretches on forever. You also saw how to generate multiples by repeated addition or multiplication, how to test whether a given number is a multiple by dividing and checking for a zero remainder, and — most importantly — how factors and multiples are two sides of the same coin: if $A$ is a factor of $B$, then $B$ is a multiple of $A$.

Up next, you will put these ideas to work in a set of hands-on exercises. You will build bus departure schedules from route intervals, judge multiple relationships on the fly, translate between factor and multiple language, and explain a real-world delivery scenario in your own words. Jump in and see how naturally factors and multiples start speaking the same language!