Evaluating modulos with large powers I need some help evaluating: $$13^{200} (mod \ 6)$$
What I've been trying to do:
$$13^1 \equiv 1 (mod \ 6)$$
$$13^2 \equiv 1 (mod \ 6)$$
Can I just say that: $$13^{200} = 13^2 * 13^2 * ... * 13^2 \equiv 1^{200}$$
Or is this incorrect?
Thanks in advance.

Since $gcd(13,6)=1$ we apply Euler's Theorem:
$$13^{\phi{(6)}} \equiv 1 \pmod 6 \Rightarrow 13^2 \equiv 1 \pmod 6$$
$$13^{200} \equiv (13^2)^{100} \equiv 1^{100} \equiv 1 \pmod 6$$
