Question: prove that n-1 is divisible by 3?
Answer: Sub in 3k = 4^n -1 to get 4*(3*k + 1) -1 = 12 *k + 4 - 1 = 12 *k +3 = 3*(4*k + 1). 4*k + 1 is an integer since k is an integer, so 4^(n+1) -1 is divisible by 3. ... For any positive integers a and n, a^n-1 is divisible by (a-1), and we can prove this using modular arithmetic.