Question

Theory of Numbers: If n is element of Z, prove that (4^n - 1) is divisible by 3.

Answer 1

This took me awhile. I used mathematical induction.
So we prove it works for 1: 4^1-1=3 which is divisible by 3.
Then we assume it works for 4^n-1 and check if it then works for 4^(n+1)-1
So 4^(n+1)-1=4(4^n-1)+3
Since we assume 4^n-1 is divisible by 3, lets write it as 3a
So 4(3a)+3. 3(4a+1). This is a factor of 3 and therefore divisible by 3

Answer 2

Well we assume it works or n and check if it then works for (n+1)