Last digit in base $10$ for $3^{22}$?

Question

Find the last digit (in base 10) for the number: $$ {3}^{22}$$

This is a question in my course in discrete math. I have checked my notes from the lectures but can't seem to solve this one.

Answer 1

Hint: $3^2=9=−1\bmod10$

Is that enough? I can add more ...

So $3^{22}=(-1)^{11}=-1=9\bmod10$.

Check: $3^{22}=31381059609$

Answer 2

We have

$$3^4=81\equiv1\bmod10 \implies 3^{22}\equiv3^{2}\bmod10$$

Answer 3

If you do not know modular arithmetic you may use the binomial formula

$$3^{22} = (10-1)^{11} = 10\cdot \underbrace{m}_{m \in \mathbb{N}} + (- 1)^{11}= 10(m-1)+ \boxed{9}$$

So, $9$ is the last digit in base $10$.