Number systems - exercises

Prove that 2 is not rational, as follows. Suppose otherwise and try to get a contradiction. So suppose that 2=(

)² for some integers . Let be least such that there are with 2=()². Observe that 2.²= and use this to deduce are both even and hence there are smaller with 2=( ' )².

Use the idea of the proof that 2 is not rational to show that, for any positive integer , either is an integer or else is not rational.