Question

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

Answer 1

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).