Prime Factorization of the Denominator

Introduction 

In number theory, the prime factors of a number are the set of prime numbers which can divide the number without a remainder. Finding the prime number factors of a number is called as prime factorization.

For example, the prime factorization of 15 is 3, 5. Here, 3 and 5 are prime numbers and the product of 2 and 5 gives 10.

Prime factorization of the denominator is mainly used for reducing fractions. If the numerator and denominator of a fraction has common terms, then to reduce the fraction prime factorization of numerator and denominator are used.

Prime Factorization of the Denominator

Here we will see an example for the prime factorization of the denominator.

To reduce the fraction, `16/26`first we have to find the prime factorization of the numerator and the denominator.

Prime factorization of the numerator (16) = 2 * 2 * 2 * 2

Prime factorization of the denominator (26) = 2 * 13

Now the fraction can be written as,

`16/26` = `(2 xx 2 xx 2 xx 2)/(2 xx 13)`

= `(2 xx 2 xx 2)/13`

= `8/13`

So, `8/13` is the reduced form of `16/26`

Example Problems on Prime Factorization of the Denominator

Here we will see some example problems on prime factorization of the denominator.

Example 1

Reduce the fraction, `152/456`

Solution

To reduce the given fraction, first we have to find the prime factorization of the numerator and the denominator.

Prime factorization of the numerator (152) = 2 * 2 * 2 * 19

Prime factorization of the denominator (456) = 2 * 2 * 2 * 3 * 19

So, the given fraction can be written as,

`152/456` = `(2 xx 2 xx 2 xx 19)/(2 xx 2 xx 2 xx 3 xx 19)`

= `1/3`

So, `1/3` is the reduced form of `152/456`

Example 2

Reduce the fraction, `12/46`

Solution

To reduce the given fraction, first we have to find the prime factorization of the numerator and the denominator.

Prime factorization of the numerator (12) = 2 * 2 * 3

Prime factorization of the denominator (46) = 2 * 23

So, the given fraction can be written as,

`12/46` = `"(2 xx 2 xx 3)/(2 xx 23)`

= `(2 xx 3)/23`

= `6/23`

So, `6/23` is the reduced form of `12/46`