Given in , we say that m is even if in , and m is odd otherwise.(a) Show that if p is odd,...

Given in , we say that m is even if in , and m is odd otherwise.

(a) Show that if p is odd, m = 2n + 1 for some . [Hint: Choose n so that n<m/2<n + 1.]

(b) Show that if p and q are odd, so are p ∙ q and pn, for any .

(c) Show that if a > 0 is rational, then a = m/n for some in,  where not both m and n are even. [Hint: Let n be the smallest element of the set .]

(d) Theorem.  is irrational.