## PhoenixFire 3 years ago Prove that if n-2 is divisible by 4 then n^2 - 4 is divisible by 16.

1. PhoenixFire

For a given integer.

2. anonymous

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

3. zzr0ck3r

hehe I wish, sec

4. zzr0ck3r

5. PhoenixFire

$\forall{n}\in \mathbb{Z} : 4|n-2 \rightarrow 16|n^2-4$ I believe that's the correct notation.

6. zzr0ck3r

yeah

7. zzr0ck3r

do you need to show for all n?

8. zzr0ck3r

nm i c

9. PhoenixFire

I need to show the proof.

10. zzr0ck3r

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

11. zzr0ck3r

sorry direct proof was fast I think

12. PhoenixFire

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.

13. zzr0ck3r

yeah 16 | (16* any integer)

14. PhoenixFire

Thanks for the help.

15. zzr0ck3r

np