See 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 ##sqrt(15) = p/q## for some ##p, q in NN##. and that ##p## and ##q## are the smallest such positive integers.

Then ##p^2 = 15 q^2##

The right hand side has factors of ##3## and ##5##, so ##p^2## must be divisible by ##3## and by ##5##. By the unique prime factorisation theorem, ##p## must also be divisible by ##3## and ##5##.

So ##p = 3 * 5 * k = 15k## for some ##k in NN##.

Then we have:

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

Divide both ends by ##15## to find:

##q^2 = 15 k^2##

So ##15 = q^2 / k^2## and ##sqrt(15) = q/k##

Now ##k < q < p## contradicting our assertion that ##p, q## is the smallest pair of values such that ##sqrt(15) = p/q##.

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