Prove that if $a \equiv_4 b$ then $3^a \equiv_{10} 3^b$

Question

Let $a,b \in \mathbb{Z}$. How can I prove that $a \equiv_4 b \rightarrow 3^a \equiv_{10} 3^b$ with basic number theory?

Answer 1

Hint: $3^{4+n}=3^4\cdot3^n\equiv_{10}3^n$.

Answer 2

Apply Euler's theorem. (notice that 3 and 10 are coprimes plus $\varphi(10)=4$.)

Answer 3

Note that from $$a \equiv b \mod (4) $$ we get $a=4k+b$

Thus $$3^a-3^b = 3^{4k+b}-3^b $$

$$=3^b ( 81^k-1)\equiv 0 \mod (10)$$

Answer 4

$\phi(10)=4$, so $x^4\equiv_{10}1\iff\gcd(x,10)=1$. Now, $a-b\equiv_4 0$, so $$3^{a-b}\equiv_{10}1\iff 3^a\equiv_{10}3^b$$