Question

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

Answer 1

For a given integer.

Answer 2

Say that n is 26 so 26-2=24 and 24 is divisible by 4. 26x 2-4=52-4=48 and 48 is divisible by 16

Answer 3

hehe I wish, sec

Answer 4

I would do contradiction

Answer 5

\[\forall{n}\in \mathbb{Z} : 4|n-2 \rightarrow 16|n^2-4\]

I believe that's the correct notation.

Answer 6

yeah

Answer 7

do you need to show for all n?

Answer 8

nm i c

Answer 9

I need to show the proof.

Answer 10

ok assume n-2= 4k for some k in Z
then n = 4k+2
then n^2-4 = (4k+2)^2 - 4 = 16k^2 + 16k +4-4 = 16(k^2+k)
since k^2+k is in Z 16|n^2-4

Answer 11

sorry direct proof was fast I think

Answer 12

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.

Answer 13

yeah 16 | (16* any integer)

Answer 14

Thanks for the help.

Answer 15

np