Fatores comuns

Common Factors

Factors are whole numbers that can be multiplied together to produce another whole number.

When solving for the factors of two or more numbers, we might encounter factors that are the same. These are called common factors. They're also sometimes called common divisors.

Let’s take a look at an example. What are the common factors of $10$ and $25$?

First, solve for the factors of $10$.

The factors of $10$ are: $$1, 2, 5, 10$$ Next, solve for the factors of $25$. The factors of $25$ are: $$1, 5, 25$$ The factors common to both lists are: $$1, 5$$ Therefore, the common factors of $10$ and $25$ are $1$ and $5$.

We could also look for prime factorizations for $10$ and $25$. In this case we can also try finding the greatest common factor. Finding prime numbers which are factors of our original numbers will be helpful.

We are now going to take a look at some of the types of problems you might encounter about finding common factors.

First, solve for the factors of $12$.

The factors of $12$ are: $$1, 2, 3, 4, 6, 12$$ Next, solve for the factors of $30$. The factors of $30$ are: $$1, 2, 3, 5, 6, 10, 15, 30$$ The factors common to both lists are: $$1, 2, 3, 6$$ Therefore, the common factors of $12$ and $30$ are $1, 2, 3,$ and $6$.

First, solve for the factors of $24$. The factors of $24$ are: $$1, 2, 3, 4, 6, 8, 12, 24$$ Next, solve for the factors of $56$. The factors of $56$ are: $$1, 2, 4, 7, 8, 14, 28, 46$$ The factors common to both lists are: $$1, 2, 4, 8$$ $24$ and $56$ have $4$ common factors.

1. Let's look at the first pair of numbers: $17$ and $33$. $\\$ The factors of $17$ are: $1$ and $17$. $\\$ The factors of $33$ are: $1$ and $33$. $\\$ The only common factor is $1$, and $1$ is not greater than $1$. $\\$
2. Look at the second pair of numbers: $15$ and $21$ $\\$ The factors of $15$ are: $1$, $3$, $5$, $15$. $\\$ The factors of $21$ are: $1, 3, 7, 21$. $\\$ The common factors are $1$ and $3$, and since $3$ is greater than $1$, it is an answer to this problem.
3. The next pair of numbers are $12$ and $31$. $\\$ The factors of $12$ are: $1, 2, 3, 4, 6, 12$. $\\$ The factors of $31$ are: $1, 31$. $\\$ The only common factor is $1$, and $1$ is not greater than $1$. $\\$
4. The fourth pair of numbers are $32$ and $48$. $\\$ The factors of $32$ are: $1, 2, 4, 8, 16, 32$. $\\$ The factors of $48$ are: $1, 2, 3, 4, 6, 8, 12, 16, 24, 48$. $\\$ The common factors are $1, 2, 3, 4, 8, 16$, and since most of the common factors are greater than $1$, it is an answer to the problem.
5. Lastly, look at the pair $11$ and $71$. $\\$ The factors of $11$ are: $1, 11$. $\\$ The factors of $71$ are: $1, 71$. $\\$ The only common factor is $1$, and $1$ is not greater than $1$. $\\$

First, solve for the factors of $30$. The factors of $30$ are: $$1, 2, 3, 5, 6, 10, 15, 30$$ Next, solve for the factors of $50$. The factors of $50$ are: $$1, 2, 5, 10, 25, 50$$ The factors common to both lists are: $$1, 2, 5, 10$$ The sum of the common factors are: $$\begin{align} 1+2+5+10= \boxed{18} \\ \end{align}$$

What are common factors?

Before getting into common factors, let's recall what factors are.

Let's say we have some number $n$. For a number to be a factor of $n$, $n$ must be divisible by this number.

Factors are whole numbers that multiply to get another number.

For example, what are the factors of $12$?

Common factors or two numbers are the factors the two (or more) numbers have in common.

For example, what are the common factors of $14$ and $21$?

We list all the factors of each number:

Identify the factors the two numbers have in common:

Therefore, the common factors of $14$ and $21$ are:

Here is a graphic explaining common factors and giving a few more examples.

Finally, the Greatest Common Factor is the largest of the factors we found. It is sometimes abbreviated GCF.

Let's do one more example!

(Video) Finding Common Factors

by TeacherTube Math

Here's a video by TeacherTube Math showing you how to find the common factors of two numbers.

$\large{\text{Summary}}$

Finding common factors of two numbers is simple. You list the factors of each number and then select all the ones that show up in both.

For example, if you want to find the common factors of $12$ and $24$. $$\begin{align} &12:\quad1,2,3,4,6,12\\ &24:\quad1,2,3,4,6,8,12,24 \end{align}$$

The numbers that appear in both are: $1,2,3,4,6,12$. These are the common factors.