Factor A factor of an integer is any integer that divides the given integer with no remainder.

Prime Number A prime number is a natural number greater than one that has exactly two positive factors: one and itself.

Exercise Write the prime factorization of the number. 12 22 x 3

Exercise Write the prime factorization of the number. 18 2 x 32

Exercise Write the prime factorization of the number. 15 3 x 5

Exercise List in order all the factors of 12. 1, 2, 3, 4, 6, 12

Exercise List in order all the factors of 18. 1, 2, 3, 6, 9, 18

Exercise List in order all the factors of 15. 1, 3, 5, 15

Greatest Common Factor The greatest common factor (also known as the greatest common divisor) is the largest number that is a factor of all the given numbers.

What is the GCF of 12 and 18?

Factoring to Find the Greatest Common Factor Write the prime factorization of each number. Identify the prime factors common to all the given numbers. Find the product of the common prime factors.

Example 1 Find the GCF of 24 and 36. 24 = 2 • 2 • 2 • 3 36 = 2 • 2 • 3 • 3 GCF = 2 • 2 • 3 = 12

Example 2 Find the GCF of 180 and 600. 180 = 2 • 2 • 3 • 3 • 5 600 = 2 • 2 • 2 • 3 • 5 • 5 GCF = 2 • 2 • 3 • 5 = 60

Relatively Prime Numbers are relatively prime if their greatest common factor is one.

Example 3 Find the GCF of the three numbers 1,764; 630; and 252. 1,764 = 2 • 2 • 3 • 3 • 7 • 7 630 = 2 • 3 • 3 • 5 • 7 252 = 2 • 2 • 3 • 3 • 7 GCF = 2 • 3 • 3 • 7 = 126

4x 4x2

3ab2 5a2b

3y 4z

Example 4 Find the GCF of 15x3y and 21x2. 15x3y = 3 • 5 • x • x • x • y 21x2 = 3 • 7 • x • x GCF = 3 • x • x = 3x2

Exercise Find the GCF of the numbers given in factored form. 23 • 32 and 22 • 32 36

Exercise Find the GCF of the numbers given in factored form. 2 • 32 • 11 and 3 • 5 • 112 33

Exercise Find the GCF of the numbers given in factored form. 22 • 3 • 52 and 22 • 52 • 7 100

Exercise Find the GCF of the numbers given in factored form. 3 • 5, 22 • 5 and 23 • 3 1

Exercise Find the GCF of the numbers given in factored form. 2 • 32, 2 • 3 • 5 and 22 • 3 • 5 6

Exercise Find the GCF of the numbers given in factored form. 23 • 52 • 11, 22 • 5 • 72 and 24 • 53 • 7 20