Two words come up again and again once you start multiplying: factors and multiples. They are really two views of the same idea — one looks inside a number, the other looks onward from it.

A factor of n is a whole number that divides it exactly — no remainder. So the factors of 12 are the numbers you can split 12 into equal rows with:

A multiple of n is what you get by multiplying n by 1, 2, 3, \dots — its times-table list:

Factors make neat rectangles

Here is the picture worth keeping. If you can lay out n dots in a perfect rectangle — equal rows, no gaps, none left over — then the number of rows and the number in each row are both factors of n. Every rectangle you can build is one factor pair; a number with lots of rectangles has lots of factors.

Press play, then step through. Each step lays out one rectangle for the chosen number, and the two side lengths are a factor pair — because that side length divides the number exactly.

Two sides of the same coin

Factors and multiples are tied together by one clean rule. Saying 3 is a factor of 12 is exactly the same statement as saying 12 is a multiple of 3:

Both just mean a = b \times k for some whole number k. So you never really learn two things here — you learn one relationship and name it from whichever end you are standing at.

Khan Academy walks through finding factors and multiples here: