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anonymous
 4 years ago
Suppose a is an integer. Prove that if 32 ∤ ((a^2 + 3)(a^2 + 7)), then a is even.
anonymous
 4 years ago
Suppose a is an integer. Prove that if 32 ∤ ((a^2 + 3)(a^2 + 7)), then a is even.

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myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2Let me take a stab at this. So I'm thinking to prove the contrapositive. If a is odd, then 32((a^2+3)(a^2+7)). Let a=2k+1 since a is odd for some integer k. Now 32((a^2+3)(a^2+7)) =>there is some integer m such that 32m=(a^2+3)(a^2+7) Now remember we let a=2k+1 So we have \[32m=((2k+1)^2+3)((2k+1)^2+7)\] \[32m=(4k^2+4k+1+3)(4k+4k+1+7)\] .... This is what I'm thinking so far Still working

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2\[32m=(4k^2+4k+1+3)(4k^2+4k+1+7)\] typeo above

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2\[32m=(4k^2+4k+4)(4k^2+4k+8)\]

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2\[32m=16k^4+16k^3+32k^2+16k^3+16k^2+32k+16k^2+16k+32\] \[32m=16k^4+k^3(16+16)+k^2(32+16+16)+k(32+16)+32\] \[32m=16k^4+32k^3+64k^2+48k+32\] \[2m=k^4+2k^3+4k^2+3k+2\] Thinking...

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2So somehow we need to show \[k^4+2k^3+4k^2+3k+2\] is even for any integer

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2We could try induction!

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.032m=(4k^2+4k+4)(4k^2+4k+8) 32m=16(k^2+k+1)(k^2+k+2)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0since one of (k^2+k+1) or (k^2+k+2) is even, thus 16(k^2+k+1)(k^2+k+2) divides 32m

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2you are right pizza! :)

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0np, myininaya did most of the work

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2pizza i seen we had the same idea to do contrapositive lol

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2i don't think i could see a way to prove the statement as is

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0didn't know where to go with it, but the numbers looked too nice when you got to the 32m part

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.2sometimes i don't know when i will run into a dead end

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Anyways, Happy Birthday, XD
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