## anonymous 4 years ago Prove: Suppose a is an integer. If 32 does not divide ((a^2 + 3)*(a^2 + 7)), then a is even.

Consider any odd number, m=2k+1 $$(k \in Z)$$ then $$m^2=(2k+1)^2=4k^2+4k+1\equiv 1 \ (mod\ 4)$$ therefore $$(m^2+3)\equiv 0\ (mod\ 4)$$ and one of $$(m^2+3)\ or\ (m^2+7) \ \equiv \ 0 \ (mod 8)$$ Therefore $$32 | (m^2+3)(m^2+7)$$ if m is odd. I will leave it to you to prove that 32 does not divide if m is not odd (i.e. even).