Question

Prove that square root of 15 is an irrational number

Answer 1

Suppose √15=p.q for some p,q∈N. and that p and q are the smallest such positive integers.

Then p2=15q2

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∈N.

Then we have:

15q2=p2=(15k)2=15⋅(15k2)

Divide both ends by 15 to find:

q2=15k2

So 15=q2k2 and √15=qk

Now k<q<p contradicting our assertion that p,q is the smallest pair of values such that √15=pq.

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