2.1 Introduction and Key Insights

Before we discuss great or common factors, let's use an example to help define what a factor is all by itself. Take the number 12. We can write $12=3\times 4$, which shows that three divides into 12 with zero remainder (and that four divides into 12, too). We then say that three is a factor of 12 (and that four is a factor of 12 too). We can also write $12=2\times 6$, so two and six are also factors of 12. We can also write $12=(-3)\times (-4)$ so $-3$ and $-4$ are also factors of 12. In fact, all together, we have:

The (positive) factors of 12 are $\pm1, \pm2, \pm3, \pm4, \pm6$, and $\pm 12$

More generally, an integer $n$ is a factor of a number $m$ means that $n\times (another\:integer)=m$. Some numbers have lots of factors, while others are special because they have just a few factors. For example, the only (positive) factors of the number 11 are one and itself. We call a number that has exactly two factors, such as 11, a prime number.

The greatest common factor (GCF) for a collection of whole numbers is the largest positive integer that divides into every number in the collection. Note that when we say "divides into" we mean "with zero remainder".

For example, let's consider the numbers 12 and eight. We can divide both of those numbers by two: $\frac{12}{2}=6$ and $\frac{8}{2}=4$. But, we can also divide both of those numbers by four: $\frac{12}{4}=3$ and $\frac{8}{4}=2$. We cannot divide both numbers by any other factor. Therefore, the GCF for these numbers is four.

We can also see the fact that GCF(12, 8) = 4 by writing out all of the (positive) factors of each and noting the largest one that they share:

Here's another example. Consider the four numbers 50, 75, and 100. Can you work out that GCF(50, 75, 100) = 25?