Prove $a\cdot (10)^n\equiv a\cdot 1\pmod 3$ using induction.

Question

Prove $a\cdot (10)^n\equiv a\cdot 1\pmod 3$ using induction.

I need to prove this equation to be true not sure how to solve. I know I have to first use one and then plug in $k+1$ but what am I plugging it into a or n and then how do I solve?

Answer 1

$$a\cdot 10^k \equiv a \quad(mod3)$$ $$a\cdot 10^{k+1}=a\cdot 10^k\cdot (9+1)\equiv a\cdot 10^k \equiv a \quad(mod3)$$