Factoring Out Greatest Common Factor

Greatest Mutual Cistron

The highest number that divides exactly into ii or more numbers.
It is the "greatest" thing for simplifying fractions!

Let'southward start with an Case ...

greatest common factor

Greatest Common Gene of 12 and 16

Discover all the Factors of each number,
Circle the Common factors,
Choose the Greatest of those

Then ... what is a "Factor" ?

Factors are numbers we can multiply together to get another number:

factors

A number tin have many factors:

Factors of 12 are 1, 2, 3, four, 6 and 12 ...

... because ii × 6 = 12, or iv × 3 = 12, or i × 12 = 12.

(Read how to find All the Factors of a Number. In our case nosotros don't need the negative ones.)

What is a "Common Factor" ?

Say we take worked out the factors of two numbers:

Case: Factors of 12 and xxx

Factors of 12 are 1, 2, iii, 4, six and 12

Factors of 30 are 1, 2, 3, five, 6, 10, 15 and 30

Then the mutual factors are those that are found in both lists:

Notice that 1, 2, 3 and 6 appear in both lists?
So, the mutual factors of 12 and 30 are: one, ii, iii and 6

Information technology is a common gene when it is a factor of two (or more than) numbers.

Here is another instance with iii numbers:

Instance: The common factors of fifteen, 30 and 105

Factors of xv are 1, 3, 5, and 15

Factors of 30 are 1, ii, three, 5, 6, 10, 15 and 30

Factors of 105 are 1, three, five, vii, xv, 21, 35 and 105

The factors that are mutual to all iii numbers are 1, three, v and 15

In other words, the common factors of fifteen, 30 and 105 are i, iii, 5 and 15

What is the "Greatest Mutual Factor" ?

Information technology is but the largest of the mutual factors.

In our previous example, the largest of the mutual factors is fifteen, and so the Greatest Mutual Gene of xv, 30 and 105 is fifteen

The "Greatest Mutual Factor" is the largest of the common factors (of two or more numbers)

Why is this Useful?

One of the most useful things is when we want to simplify a fraction:

Example: How tin can we simplify 12 thirty ?

Before nosotros found that the Common Factors of 12 and 30 are ane, 2, 3 and 6, and so the Greatest Common Gene is 6.

Then the largest number we tin carve up both 12 and xxx exactly past is vi, similar this:

	÷ 6

12 xxx	=	2 5

	÷ 6

The Greatest Common Factor of 12 and 30 is 6.

So 12 30 can be simplified to ii 5

Finding the Greatest Mutual Gene

Here are three ways:

1. We tin can:

find all factors of both numbers (utilise the All Factors Calculator),
then detect the ones that are common to both, and
then choose the greatest.

Example:

Two Numbers	Factors	Common Factors	Greatest Mutual Gene	Example Simplified Fraction
ix and 12	9: i, 3, 9 12: 1, 2, 3, 4, six, 12	1, three	3	nine 12 = 3 4

And another example:

Ii Numbers	Factors	Common Factors	Greatest Common Factor	Example Simplified Fraction
6 and 18	6: ane, 2, 3, 6 18: ane, 2, 3, 6, ix, 18	1, 2, 3, 6	6	6 18 = 1 iii

2. Or we tin can notice the prime factors and combine the common ones together:

Two Numbers	Thinking ...	Greatest Common Factor	Case Simplified Fraction
24 and 108	2 × 2 × 2 × 3 = 24, and 2 × 2 × three × 3 × three = 108	2 × 2 × 3 = 12	24 108 = 2 9

3. Or sometimes we can just play around with the factors until we find it:

2 Numbers	Thinking ...	Greatest Common Factor	Example Simplified Fraction
9 and 12	3 × iii = nine and 3 × 4 = 12	3	ix 12 = iii iv

But in that case we must check that we accept found the greatest common factor.

Greatest Mutual Factor Computer

OK, there is likewise a really easy method: we can utilize the Greatest Common Factor Calculator to observe it automatically.

Other Names

The "Greatest Mutual Factor" is often abbreviated to "GCF", and is also known as:

the "Greatest Mutual Divisor (GCD)", or
the "Highest Common Factor (HCF)"