How do you prove that square root 15 is irrational?
This proof uses the unique prime factorisation theorem that every positive integer has a unique factorisation as a product of positive prime numbers.
The right hand side has factors of
Then we have:
#15 q^2 = p^2 = (15k)^2 = 15*(15 k^2)#
Divide both ends by
#q^2 = 15 k^2#
So our initial assertion was false and there is no such pair of integers.