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.

Try looking at the last digits for small powers of $3$. See a pattern? — Rushabh Mehta, Sep 25 '19 at 13:04

almagest · Accepted Answer · 2019-09-25T08:02:54.560

4

Hint: $3^2=9=−1\bmod10$

Is that enough? I can add more ...

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

Check: $3^{22}=31381059609$

edited Sep 25 '19 at 08:02

answered Sep 25 '19 at 07:50

almagest

score 1 · Answer 2 · answered Sep 25 '19 at 08:10

1

We have

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

answered Sep 25 '19 at 08:10

user

score 1 · Answer 3 · answered Sep 25 '19 at 08:48

1

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 $10$.

answered Sep 25 '19 at 08:48

trancelocation

3 Answers3