1. ## Mod Question

Show that 5^n + 6^n = 0 mod 11, for all odd number n

My initial attempt was to prove by induction. Base case is obvious, and the induction step would be to prove for n + 2.

I'm trying to play around by writing in the mod definition but no luck so far. Any advice?

2. ## Re: Mod Question

Originally Posted by hashshashin715
Show that 5^n + 6^n = 0 mod 11, for all odd number n

My initial attempt was to prove by induction. Base case is obvious, and the induction step would be to prove for n + 2.

I'm trying to play around by writing in the mod definition but no luck so far. Any advice?
You want to show that IF

$5^k+6^k$ is divisible by 11

THEN

$5^{k+2}+6^{k+2}$

will also be divisible by 11.

So

$5^{k+2}+6^{k+2}=(25)5^k+(36)6^k=(22)5^k+(33)6^k+3 \left(5^k+6^k\right)$

3. ## Re: Mod Question

Thanks a lot, i got it now.

I was getting stuck at the part where we have (25)5^k + (36)5^k. Breaking them down like that was clever.