# 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?

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