PRIME FACTORIZATION

To simplify fractions we reduce by common factors. One way to reduce fractions is to find the prime factorization of the numerator and denominator and reduce by the common factors.

Factorization: An expression of a number as a product of factors.

Example: 18 = 2 x 9.

Prime factorization: An expression of a number as a product of prime factors.

Example: 18 = 2x3x3.

Example: Simplify 18/24.

Solution:

We can use divisibility rules to simplify factorization.

Example: Find the prime factorization of 1440.

Solution (Method 1): We note that 1440 is divisible by 10, so we can factor 10 out of 1440:

1440 = 10 x 144.

Now we use the fact that 144 is divisible by 9 (4+4+1=9, and 9 is divisible by 9):

144 = 9 x 16.

Therefore, 1440 = 9 x 16 x 10 

= (3x3)x(2x2x2x2)x(2x5)

= .

This problem is illustrated below with a factor tree. The prime factors are circled.

Solution Method 2: We can find the prime factorization more methodically by dividing the smallest prime factor into 1440, dividing the resulting quotient by the next smallest prime factor, and so on.

Note that we can use each quotient as the divisor for the next division. The prime factors are the divisors and the last quotient:

1440 = (2)(2)(2)(2)(2)(3)(3)(5)

=.