maior divisor comum

Greatest Common Factor

You've already learned about common factors. The Greatest Common Factor (also called the GCF, Greatest Common Divisor, or GCD) of two integers is a very similar concept!

Think about it. The GCF of two integers is the Greatest (largest) Common (shared) Factor (divisor).

Let's take a look at one of the examples from the graphic.

Greatest Common Factor of $28$ and $36$

We first list the factors of each integer. $$\begin{align} \text{factors of } 28&: 1, 2, 4, 7, 14, 28\\ \text{factors of } 36&: 1, 2, 3, 4, 6, 9, 12, 18, 36 \end{align}$$ Now, we identify the factors they share, which are boxed below. $$\begin{align} \text{factors of } 28&: \boxed{1}, \boxed{2}, \boxed{4}, 7, 14, 28\\ \text{factors of } 36&: \boxed{1}, \boxed{2}, 3, \boxed{4}, 6, 9, 12, 18, 36 \end{align}$$ Finally, the greatest common factor of the two integers is the largest of these shared factors, boxed below. $$\begin{align} \text{factors of } 28&: 1, 2, \boxed{4}, 7, 14, 28\\ \text{factors of } 36&: 1, 2, 3, \boxed{4}, 6, 9, 12, 18, 36 \end{align}$$

Here are some other methods of finding the greatest common factor, in case this one is too tedious.

Let's do another example to practice!

First, we list the factors of each integer. $$\begin{align} \text{factors of } 16&: 1, 2, 4, 8, 16\\ \text{factors of } 48&: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 \end{align}$$ Now, let's identify the factors they share, which are boxed below. $$\begin{align} \text{factors of } 16&: \boxed{1}, \boxed{2}, \boxed{4}, \boxed{8}, \boxed{16}\\ \text{factors of } 48&: \boxed{1}, \boxed{2}, 3, \boxed{4}, 6, \boxed{8}, \boxed{16}, 24, 48 \end{align}$$ Finally, the greatest common factor of the two integers is the largest of these shared factors, boxed below. $$\begin{align} \text{factors of } 16&: 1, 2, 4, 8, \boxed{16}\\ \text{factors of } 48&: 1, 2, 3, 4, 6, 8, 12, \boxed{16}, 24, 48 \end{align}$$

So, the greatest common factor of $16$ and $48$ is $16.$

(Video) 3 Methods for Greatest Common Factor

by Mathademics!

This video by Mathademics! goes over three different methods for finding the Greatest Common Factor (also called the GCF, Greatest Common Divisor, or GCD).

$\large{\text{Summary}}$

Method 1: List the Factors

The first method is very simple. Just list the factors out and select the largest one that they share.

Let's find the greatest common factor of $28$ and $36$ using this method. $$\begin{array}{cl} 28&{\color{#29A603}{1}},{\color{#29A603}{2}},{\color{#29A603}{4}},7,28\\ 36&{\color{#29A603}{1}},{\color{#29A603}{2}},3,{\color{#29A603}{4}},6,9,12,18,36 \end{array}$$ Out of the three factors they have in common, $4$ is the greatest!

Method 2: Prime Factorization

The second method is similar to the first. We want to list the prime factors of each number. Then, we select the prime factors that they have in common. Multiplying these together gives us the greatest common factor.

Let's find the greatest common factor of $28$ and $36$ using this method. $$\begin{array}{cl} 28&{\color{#29A603}{2}}\cdot{\color{#29A603}{2}}\cdot7\\ 36&{\color{#29A603}{2}}\cdot{\color{#29A603}{2}}\cdot3\cdot3 \end{array}$$ We have two prime factors that are in each: $2$ and $2$. Multiplying these together we get $2\times2=4$ which is our greatest common factor!

Method 3: Upside-Down Division

The third method is probably the most annoying. To be fair I have to draw out diagrams so that's what makes it annoying. You want to divide both numbers by common primes until they both have prime numbers below. Then you multiply the numbers along the side together.

Let's find the greatest common factor of $28$ and $36$ using this method.

Since we can't factor anything other than $1$ into $7$ and $9$, we know we're done. Finally, multiply the primes on the left together: $2\times2=4$

(Video) Greatest Common Factor

by tecmath

This video by tecmath shows a quick, simple way to find the greatest common factor, or GCF.

$\large{\text{Summary}}$

Suppose we have the numbers $60$ and $72$. We look at these two numbers and ask, "Is there any number that goes into both of these?" The answer is $2$! So we divide each of them by $2$. $$\begin{array}{ccc} &60&72\\ \hline \color{#29A603}{2}&30&36 \end{array}$$ Now, we ask the question again. $2$ goes into each of these as well. $$\begin{array}{ccc} &60&72\\ \hline \color{#29A603}{2}&30&36\\ \color{#29A603}{2}&15&18\\ \end{array}$$ What about these next numbers? You might notice that $3$ goes into each of these. $$\begin{array}{ccc} &60&72\\ \hline \color{#29A603}{2}&30&36\\ \color{#29A603}{2}&15&18\\ \color{#29A603}{3}&5&6\\ \end{array}$$ There's now no number (besides $1$) that goes into each of these. So our final step is to multiply the three numbers along the side. $$2\cdot2\cdot3=12$$ So our greatest common factor is $12$.

Try it out on your own!