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?

have you tried Euler's theorem? (https://en.wikipedia.org/wiki/Euler%27s_theorem) — Lacramioara, Nov 05 '18 at 17:25

score 2 · Answer 1 · answered Nov 05 '18 at 17:32

2

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

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

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

answered Nov 05 '18 at 17:32

Thank you! But i don't see why $$3^b ( 81^k-1)\equiv_{10} 0 $$ holds ? – kerlifa Nov 05 '18 at 18:00
$81^k$ ends up with $1$ therefore $81^k-1$ ends up with $0$ which makes it a multiple of $10$ – Mohammad Riazi-Kermani Nov 05 '18 at 18:06

score 1 · Answer 2 · answered Nov 05 '18 at 17:31

1

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

answered Nov 05 '18 at 17:31

Lacramioara

score 0 · Answer 3 · answered Nov 05 '18 at 17:26

0

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

answered Nov 05 '18 at 17:26

Arthur

score 0 · Answer 4 · answered Nov 05 '18 at 17:36

0

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

answered Nov 05 '18 at 17:36

4 Answers4