PhoenixFire
Prove that if n2 is divisible by 4 then n^2  4 is divisible by 16.



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PhoenixFire
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For a given integer.

carl51
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Say that n is 26 so 262=24 and 24 is divisible by 4. 26x 24=524=48 and 48 is divisible by 16

zzr0ck3r
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hehe I wish, sec

zzr0ck3r
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I would do contradiction

PhoenixFire
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\[\forall{n}\in \mathbb{Z} : 4n2 \rightarrow 16n^24\]
I believe that's the correct notation.

zzr0ck3r
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yeah

zzr0ck3r
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do you need to show for all n?

zzr0ck3r
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nm i c

PhoenixFire
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I need to show the proof.

zzr0ck3r
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ok assume n2= 4k for some k in Z
then n = 4k+2
then n^24 = (4k+2)^2  4 = 16k^2 + 16k +44 = 16(k^2+k)
since k^2+k is in Z 16n^24

zzr0ck3r
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sorry direct proof was fast I think

PhoenixFire
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Yeah, they wanted Direct Proof.
so since n^2  4 = 16k the (k^2+k) in 16(k^2+k) doesn't matter, the rest match.
that's what was confusing me.

zzr0ck3r
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yeah 16  (16* any integer)

PhoenixFire
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Thanks for the help.

zzr0ck3r
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np