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The **greatest common divisor** (**GCD**) of a number of integers is the greatest number that all the integers can be divided by without getting a fractional part. For example, the greatest common divisor of 6 and 9 is 3 because 3 is the greatest number that divides both 6 and 9.

The greatest common divisor can be used to simplify fractions. To reduce a fraction as much as possible the numerator and denominator can be divided by the greatest common divisor for all terms in the numerator and denominator.

The fraction 6 ⁄ 9 can in this way be simplified by dividing the numerator and denominator by GCD(6, 9) = 3.

6

9

=
9

6 / 3

9 / 3

=
9 / 3

2

3

3

This works even for fractions that contain more complicated expressions. The fraction (2*x* + 8*y*) ⁄ 4 can be simplified by dividing by GCD(2, 8, 4) = 2.

2*x* + 8*y*

4

=
4

(2*x* + 8*y*) / 2

4 / 2

=
4 / 2

2

Another possible use case is when doing factorizations of polynomial expressions. The expression 8*x*^{2} + 12*x* can be factorized using the factor 4 because GCD(8, 12) = 4.

8*x*^{2} + 12*x*
=
4 × *x*^{2} + 3*x*)

8*x*^{2} + 12*x*

4_{}

=
4 × (24

Calculate the lowest common denominator for two or more fractions.