Greatest Common Factor

Factors that are shared by two or more numbers are called common factors. The greatest of these common factors is called the greatest common factor (GCF). Consider, for example, the factors of

4

and

8 .

Factors of 4 : Factors of 8 : 1, 2, 4 1, 2, 4, 8

The common factors of

4

and

8

are

1,

2,

and

4 .

The

GCF (4, 8) = 4 .

The GCF can also be determined by multiplying the prime factors shared in the prime factorization of the numbers in question.

Prime factorization of 4: 2 * 2. Prime factorization of 8: 2 * 2 * 2. The common prime factors (2 and 2) are shown within a circle. The product of these common prime factors is 4 (2 * 2).

Common factors can be extended to expressions with variable terms. For example, consider the expressions

4 x

and

2 x .

Factors of 4 x : Factors of 2 x : 1, 2, 4, x, 2 x, 4 x 1, 2, x, 2 x

The

GCF (4 x, 2 x) = 2 x .

It should be noted that the greatest common factor is also called the greatest common divisor because a factor of a number divides that number evenly.

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