Let n be a positive integer and not a perfect square. Prove √n is irrational.
Consider proving by contradiction. If √n is rational, then there exist two coprime integers p,q such that √n=pq, which implies p2=nq2.
Moreover, since p,q are coprime, by Bézout's theorem, there exist two integers a,b such that ap+bq=1.
Thus
p=ap2+bpq=anq2+bpq=(anq+bp)q, which implies √n=pq=anq+bp∈N+, which contradicts.
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