Induction, or am I missing something?

Question

I need to show that 4 is a divisor of (7^n) - (3^n) for every positive integer n. I think I need to use induction...
So far I have that when n = 1, 7-3 = 4 and 4|4 so the base case is true..
Then I assume that when n = k 4|[(7^k)-(3^k)] is true.
Then for n = k+1 I have (7^k)(7) - (3^k)(3) and I said that since 4|7-3 and also since 4|[(7^k) - (3^k)] then 4|[7^(k+1) - 3^(k+1)].

But is this correct? Or should I be stating it a little different?

Thanks!

Answer 1

You could use a modular approach: since 7^n/4=3^n mod 4 this reduces the problem to (3^n-3^n) mod 4 which is definitely 0--meaning that for any n, 4 will be a factor.

Answer 2

Simply take the equation in mod 4.
7^n-3^n := 3^n-3^n = 0 (mod 4)

Answer 3

Yes, that is correct. But you're not finished with your proof, you now have to write, without the stuff in the brackets.
"Because it is true for n=1, hence it must be true for n=2 (as you have proven it is true for n = k+1), and it must be true for n=3 etc. Therefore, it is true for all positive integers n. "