Prove that $a + 2b \equiv 0 \pmod{3}$ if and only if $a \equiv b\pmod{3}$.

Question

I need to prove that $a + 2b \equiv 0\pmod{3}$ if and only if $a \equiv b \pmod{3}$.

I know that you need to show both cases but my professor said that we weren't supposed to use one to solve the other so I'm stuck.

Answer 1

Hint: $\ 2\equiv -1\pmod3\ $ so $\ a+2b\equiv a-b\pmod 3\ $

Answer 2

Hint: Try converting the second equation to $$ a -b \equiv 0 \pmod 3 $$ by subtracting $b$ from both sides.

Answer 3

$a \equiv b \pmod 3 \iff 3|a-b \iff 3|a-b + 3b \iff 3|a+2b \iff a+2b \equiv 0 \pmod 3$

or...

$2\equiv -1 \pmod 3$ so....

$a+2b \equiv 0\pmod 3 \iff a- b \equiv 0 \pmod 3 \iff a \equiv b \pmod 3$

Answer 4

Hint: Here $2$ is "a multiple of three away from" $-1$, meaning that they are what modulo three?

They are equivalent.