# How do you prove that square root 15 is irrational?

##### 1 Answer

See explanation...

#### Explanation:

This proof uses the unique prime factorisation theorem that every positive integer has a unique factorisation as a product of positive prime numbers.

Suppose

Then

The right hand side has factors of

So

Then we have:

#15 q^2 = p^2 = (15k)^2 = 15*(15 k^2)#

Divide both ends by

#q^2 = 15 k^2#

So

Now

So our initial assertion was false and there is no such pair of integers.