Using Mods to show something is divisible by 3

Question

How do you show $5^{2 \cdot 3^k}-5^{3^k}+1$ is divisible by $3$ using mods?

I tried to simplify to $2^{2 \cdot 3^k}-2^{3^k}+1$, but now I am stuck.

Answer 1

Hints:

$5\equiv2\equiv-1$, so $5^{2\cdot3^k}-5^{3^k}+1\equiv2^{2\cdot3^k}-2^{3^k}+1\equiv(-1)^{2\cdot3^k}-(-1)^{3^{k}}+1\pmod 3 ,$

and remember that $-1$ to an even power is $1$ and to an odd power is $-1$.

Answer 2

A variant:

\begin{align} 5^{2\cdot 3^k}-5^{3^k}+1&=5^{3^k}\bigl(5^{3^k}-1\bigr)+1\\ &\equiv 5(5-1)+1 &&\text{by Fermat, since 3 is prime}\\ & \equiv -1\cdot 1+1 \mod 3. \end{align}