show that 11^n -4^n is divisible by 7

Question

isaiah.feynman · Answer

What could n mean?

zzr0ck3r · Answer

this is induction, n is natural

isaiah.feynman · Answer

I don't think that would work anyway, 7 is a prime. It won't divide any number perfectly.

anonymous · Answer

zzr0ck3r is right except I haven't got a clue what to do

anonymous · Answer

seems to work for n=1 and 2 so I assume it works for the rest of natural numbers

zzr0ck3r · Answer

so for sure this is true when n = 1
so suppose 7 divides $11^n-4^n$ for some $n\in \mathbb{N}$
so
$11^{n+1}-4^{n+1} = 11*11^n-4*4^n = (11^n-4^n)+7*11^n+3*11^n-3*4^n=\(11^n-4^n)+7*11^n+3(11^n-4^n) = 4(11^n-4^n)+7*11^n=4*7k+7*11^n$
surely 7 divides this number so by induction we are done.

isaiah.feynman · Answer

I really need to teach myself this topic throughly! Know any good books?

zzr0ck3r · Answer

I would just google "induction". Its really not that bad. just think of dominoes. Any standard group theory or number theory book will be sufficient.

anonymous · Answer

I slept on it and that is what was in my head this morning lol....the answer you gave