How do you calculate the module of a high raised number?

Question

I need some help solving this.

Find $3^{336549354297854} (mod 28)$

I don't really understand how to solve this type of problem. can anyone explain?

Answer 1

Start by looking for a pattern:

$3^0 \equiv 1 \pmod{28}$

$3^1 \equiv 3 \pmod{28}$

$3^2 \equiv 9 \pmod{28}$

$3^3 \equiv 27 \pmod{28}$

$3^4 \equiv 25 \pmod{28}$

$3^5 \equiv 19 \pmod{28}$

$3^6 \equiv 1 \pmod{28}$

$3^7 \equiv 3 \pmod{28}$

$3^8 \equiv 9 \pmod{28}$

So every $6^{th}$ value is the same. Your power can be written as:

$336549354297854=56091559049642\times6+\color{red}{2}$

So it will be the $3^2$=9.