# 1.05.06 Finding common denominators: Method 3, prime factorisation.

This is a more complicated method, but it always gives the lowest common denominator and so makes the subsequent work of finding the numerators of the equivalent fractions easier.

First you write each of the denominators as a multiple of its prime factors.
A prime number, is just a number that is only divisible by itself and 1. e.g. 1, 2, 3, 5, 7, 11 etc. are prime numbers.
A factor of a number, is just an integer, that can be multiplied by another integer to give the number.
e.g. 4 x 5 = 20, so both 4 and 5 are factors of 20.
5 is a prime factor, because it is also a prime number.

Example. 3/12 + 1/6 + 1/4
(you can probably spot that 12 is a common denominator, but we will prove this by writing each denominator as a multiple of its prime factors.)

• The prime factors which multiply to give 12 are, 1 x 2 x 2 x 3.
• (i.e. 1 x 2 x 2 x 3 = 12, and each of the numbers 1, 2, & 3, are prime numbers).
• The prime factors which multiply to give 6 are, 1 x 2 x 3.
• The prime factors which multiply to give 4 are, 1 x 2 x 2.

From above we can see that:

• When 12 is expressed as a multiple of its prime factors there is, one 1, two 2s and one 3 .
• When 6 is expressed as a multiple of its prime factors there is, one 1, one 2 and one 3 .
• When 4 is expressed as a multiple of its prime factors there is, one 1 and two 2s .

Next we identify the greatest number of times that each prime factor appears, for any of the denominators.

• The largest count of 1s for any of the denominators is 1, i.e.,     1 .
• The largest count of 2s for any of the denominators is 2, i.e.,     2, 2 .
• The largest count of 3s for any of the denominators is 1, i.e.,     3 .

Finally you multiply these numbers together to give the lowest common denominator.
This gives 1 x 2 x 2 x 3 = 12 .
i.e. the lowest common denominator is 12.

Example.

• 5/12 - 2/10 .
• 1 x 2 x 2 x 3 = 12 .
• 1 x 2 x 5 = 10 .
• largest count of 1s = one,     1 .
• largest count of 2s = two,     2 2 .
• largest count of 3s = one,     3 .
• largest count of 5s = one,     5 .
• 1 x 2 x 2 x 3 x 5 = 60 .
• 60 is the common denominator .

The calculation with equivalent fractions would be     25/60 - 12/60 = 13/60 .
Therefore     5/12 - 2/10 = 13/60 .
The fraction 13/60 is already in its lowest form, (there is no number that divides evenly into both the numerator and denominator).