# Mod Question

• Oct 5th 2011, 03:27 PM
hashshashin715
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?
• Oct 5th 2011, 03:36 PM
Re: Mod Question
Quote:

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

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

THEN

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

will also be divisible by 11.

So

$\displaystyle 5^{k+2}+6^{k+2}=(25)5^k+(36)6^k=(22)5^k+(33)6^k+3 \left(5^k+6^k\right)$
• Oct 5th 2011, 03:49 PM
hashshashin715
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.