# 初級代數: 最大公因數

1.
$$12,30$$
答案
2.
$$20,30,100$$
答案
3.
$$16,8,36$$
答案
4.
$$4,8,14$$
答案
5.
$$8,12$$
答案
6.
$$15,25,40$$
答案

# Greatest Common Factor - Introduction

All whole numbers have factors — numbers that can be multiplied together to yield the original value. For example, both
$$2$$
and
$$3$$
are factors of
$$6$$
, because
$$2\times 3=6$$
. Some numbers, such as prime numbers, have only two factors: The number itself and
$$1$$
. Consider the case of
$$17$$
— no whole numbers multiply together to give this answer, except for
$$17$$
and
$$1$$
.
Many numbers have common factors, which are factors that appear in both of their lists of factors. Let’s look at the factors of
$$8$$
and
$$10$$
, and see if they have any factors in common.
Factors of
$$8$$
$$1$$
,
$$2$$
,
$$4$$
Factors of
$$10$$
$$1$$
,
$$2$$
,
$$5$$
Looking at the lists, we can see that their common factors are
$$1$$
and
$$2$$
.

# Finding the Greatest Common Factor

Listing the factors also helps us find the greatest common factor (GCF) — the largest whole number factor that the given numbers have in common. This is especially useful when simplifying fractions. Because, once we found the GCF, we can divide both the numerator and denominator by the GCF, which will result in a simpler fraction.
How do you find the greatest common factor? Below are two methods.

## Method 1: Listing Factors

Let’s find the GCF of these numbers:
$$16$$
,
$$28$$
and
$$52$$
.
Factors of
$$16$$
$$1$$
,
$$2$$
,
$$4$$
,
$$8$$
Factors of
$$28$$
$$1$$
,
$$2$$
,
$$4$$
,
$$7$$
,
$$14$$
Factors of
$$52$$
$$1$$
,
$$2$$
,
$$4$$
,
$$13$$
,
$$26$$
Looking at the list, we can see that the greatest common factor is
$$4$$
.

## Method 2: Prime Factors

You can also find the GCF using prime factors. Let’s try it for
$$12$$
and
$$16$$
.
Prime factors of
$$12$$
$$1$$
,
$$2$$
,
$$2$$
,
$$3$$
Prime factors of
$$16$$
$$1$$
,
$$2$$
,
$$2$$
,
$$2$$
,
$$2$$
,
$$2$$
Find the “intersection” of these primes — the factors both numbers have in common — to get your answer. In this case, that’s
$$2\times 2=4$$
, which is the GCF of
$$12$$
and
$$16$$
.

# What's Next

Once you’re familiar with finding the GCF, you can use it to quickly simplify algebra equations and reduce complicated fractions. Need a quick answer for more complex factor questions? Try our greatest common factor calculator.
Ready to dive in and do more practice problems yourself? Start with Cymath’s free online practice problems or join Cymath Plus for an ad-free experience complete with more in-depth math assistance.