<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] Use mod $4$, meaning if $a = 1 \pmod 4\implies a^2+3= 0\pmod 4, a^2+7 = 0\pmod 8\implies (a^2+3)(a^2+7)= 0 \pmod {32}$ . If $a = 3\pmod 4\implies a^2+3 = 0\pmod 4, a^2+7 = 0\pmod 8\implies (a^2+3)(a^2+7) = 0\pmod {32}$ . Either case gives the answer.
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