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In order to find the greatest common factor we use primenumbers. A prime number is an integer that is greater than oneand has no factors other than itself and one.

Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.. .

## I. Factoring A Number Into Primes

1. Check to see if the first prime number, 2, divides evenlyinto the given number.

2. If it doesn't divide evenly, try the next prime number, 3.Continue until you find a prime factor.

3. Rewrite the given number as a product of the prime factorand the result from division.

4. Repeat steps 1-3 on the number resulting from division.

5. Repeat steps 1-4 until the given number is written as aproduct of primes.

Example:

 1260 1260 2 * 630 2 * 2 * 315 2 * 2 * 3 * 105 2 * 2 * 3 * 3 * 35 2 * 2 * 3 * 3 * 5 * 7

The prime factorization of 1260 is 2 * 2 * 3 * 3 * 5 * 7,which are the numbers along the left side in the above divisions.

## II. Finding The Greatest Common Factor

1. Factor each number completely into primes.

2. Look at the common factors for the two numbers.

3. Multiply the common factors together to get the GCF.

Example:

 252 90 2 * 126 2 * 45 2 * 2 * 63 2 * 3 * 15 2 * 2 * 3 * 21 2 * 3 * 3 * 5 2 * 2 * 3 * 3 * 7 2 * 2 * 3 * 3 * 7 2 * 3 * 3 * 5

The underlined factors are the ones common to both 252 and 90.

Now multiply the common factors: 2 * 3 * 3 = 18.

The greatest common factor of 252 and 90 is 18.

We can also use this method to find the GCF of twoalgebraic expressions.

Example:

 60x 2 y 210xy 2 2 * 30x 2y 2 * 105xy 2 2 * 2 * 15x 2 y 2 * 3 * 35xy 2 2 * 2 * 3 * 5 * x 2 y 2 * 3 * 5 * 7xy 2 2 * 2 * 3 * 5 * x * x * y 2 * 3 * 5 * 7 * x * y * y 2 * 2 * 3 * 5 * x * x * y 2 * 3 * 5 * 7 * x * y * y

The underlined factors are the ones common to 60x 2y and 210xy 2 .

Now multiply the common factors: 2 * 3 * 5 * x * y = 30xy.

The greatest common factor of 60x 2 y and 210xy2 is 30xy.