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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.
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.

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mathmate
 4 years ago
Best ResponseYou've already chosen the best response.0Consider 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).
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