Common Factors

Factors that two numbers have in common are called the common factors of those numbers.

Example:
What are the common factors of 20 and 25?
  The factors of 20 are 1, 2, 4, 5, and 20.
  The factors of 25 are 1, 5, and 25.
The common factors of 20 and 25 are 1 and 5.

Example:
What are the common factors of 15 and 30?
  The factors of 15 are 1, 3, 5, and 15.
  The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
The common factors of 15 and 30 are 1, 3, 5, and 15.

Example:
What are the common factors of 9 and 20?
  The factors of 9 are 1, 3, and 9.
  The factors of 20 are 1, 2, 4, 5, and 20.
The only common factor of 9 and 20 is 1.

Greatest Common Factors

The largest common factor of two numbers is called their greatest common factor or highest common factor.

Example:
What is the greatest common factor of 20 and 25?
  The common factors of 20 and 25 are 1 and 5. (see above)
The greatest common factor is 5.

Example:
What is the greatest common factor of 15 and 30?
  The common factors of 15 and 30 are 1, 3, 5, and 15.
The greatest common factor is 15.

Example:
What is the greatest common factor of 9 and 20?
  The only common factor of 9 and 20 is 1.
The greatest common factor is 1.

Method #2: Prime Factors

Another method of finding the greatest common factor uses the prime factors of each number. Rather than looking at all the factors—prime or not—to find the single highest one they have in common, we multiply all the prime factors they have in common, to reach the same results.

Example:
What is the greatest common factor of 20 and 25?
  The prime factors of 20 are 2 x 2 x 5.
  The prime factors of 25 are 5 x 5.
  The prime factor they have in common is 5.
The greatest common factor is 5.

Example:
What is the greatest common factor of 15 and 30?
  The prime factors of 15 are 3 x 5.
  The prime factors of 30 are 2 x 3 x 5.
  The prime factors they have in common are 3 x 5.
  3 x 5 = 15.
The greatest common factor is 15.

When two numbers have no prime factors in common, their greatest common factor is 1.

Example:
What is the greatest common factor of 9 and 20?
  The prime factors of 9 are 3 x 3.
  The prime factors of 20 are 2 x 2 x 5.
  9 and 20 have no prime factors in common.
The greatest common factor is 1.