<s>[INST] Prove that $32\mid(a^2+3)(a^2+7)$ If $a$ be an odd integer prove that $32\mid(a^2+3)(a^2+7)$
These are not three or two consecutive integers so they are not divisible by $3$ or $2$ also mathematical induction can not be applied. Any idea how to solve it
 [/INST] $a=2n+1 \Rightarrow a^2=4n^2+4n+1$. Then:
$$(a^2+3)(a^2+7)=(4n^2+4n+4)(4n^2+4n+8)=16(n^2+n+1)(n^2+n+2).$$
The numbers in the brackets are two consecutive numbers, so one of them must be even.
</s>