Greatest Common Factor

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

Let's offset with an Example ...

greatest common factor

Greatest Common Factor of 12 and sixteen

  1. Find all the Factors of each number,
  2. Circle the Common factors,
  3. Cull the Greatest of those

So ... what is a "Factor" ?

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

factors

A number can have many factors:

Factors of 12 are ane, ii, 3, 4, 6 and 12 ...

... because 2 × 6 = 12, or four × 3 = 12, or 1 × 12 = 12.

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

What is a "Mutual Gene" ?

Say we have worked out the factors of ii numbers:

Example: Factors of 12 and 30

Factors of 12 are ane, 2, 3, four, 6 and 12
Factors of 30 are one, 2, 3, 5, 6, 10, fifteen and 30

And then the mutual factors are those that are found in both lists:

  • Find that i, 2, 3 and 6 announced in both lists?
  • And so, the common factors of 12 and 30 are: i, ii, three and half-dozen

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

Here is another example with three numbers:

Example: The common factors of xv, 30 and 105

Factors of 15 are 1, 3, 5, and fifteen
Factors of xxx are ane, 2, 3, 5, 6, 10, 15 and 30
Factors of 105 are 1, three, 5, 7, 15, 21, 35 and 105

The factors that are common to all three numbers are i, 3, 5 and 15

In other words, the common factors of 15, 30 and 105 are ane, 3, 5 and xv

What is the "Greatest Common Cistron" ?

Information technology is simply the largest of the common factors.

In our previous example, the largest of the mutual factors is 15, then the Greatest Mutual Factor of 15, thirty and 105 is 15

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

Why is this Useful?

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

Example: How tin we simplify 12 30 ?

Before we found that the Mutual Factors of 12 and thirty are one, 2, 3 and vi, then the Greatest Common Gene is 6.

So the largest number nosotros can split both 12 and thirty exactly by is six, similar this:

÷ half-dozen
right over arrow
12 30  = 2 5
right under arrow
÷ 6

The Greatest Common Factor of 12 and 30 is six.

Then 12 30 tin can exist simplified to 2 5

Finding the Greatest Mutual Factor

Here are iii ways:

1. We can:

  • find all factors of both numbers (employ the All Factors Estimator),
  • then discover the ones that are common to both, and
  • then choose the greatest.

Example:

2 Numbers Factors Common Factors Greatest
Common Factor		 Case Simplified
Fraction
nine and 12 9: i, three, ix
12: 1, 2, 3, 4, vi, 12		 1, 3 3 9 12 = iii 4

And another example:

Two Numbers Factors Common Factors Greatest
Common Factor		 Example Simplified
Fraction
6 and eighteen 6: 1, 2, 3, half-dozen
18: 1, 2, three, 6, 9, 18		 one, 2, 3, 6 6 half-dozen xviii = 1 iii

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

Ii Numbers Thinking ... Greatest
Common Factor		 Example Simplified
Fraction
24 and 108 two × 2 × 2 × iii = 24, and
2 × 2 × 3 × 3 × 3 = 108		 2 × 2 × iii = 12 24 108 = 2 9

3. Or sometimes nosotros tin can just play around with the factors until nosotros discover it:

Two Numbers Thinking ... Greatest
Common Gene		 Example Simplified
Fraction
9 and 12 three × 3 = 9 and 3 × four = 12 iii ix 12 = iii four

But in that instance nosotros must check that we take constitute the greatest common factor.

Greatest Mutual Gene Reckoner

OK, there is also a really easy method: we can use the Greatest Mutual Factor Computer to find information technology automatically.

Other Names

The "Greatest Common Factor" is oft abbreviated to "GCF", and is as well known as:

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