Get all linear congruences of $3x\equiv 6(mod9)$

Question

So to solve this I'm told that you find the gcd of 3 and 9, which is (3,9)=3 and since 3|6, there are 3 classes of solutions which can be found using the diophantine equation $3x+9y=6$. I was only able to fine one solution which is $3(-1)+9(1)=6$ so $x=-1+9n\equiv 8 \pmod 9$. Is there a way to find the other 2 solutions? Or do I have to just keep trying values?

Answer 1

Given

$$3x\equiv 6\pmod 9 \iff3x-6=9k \iff x-2=3k \iff x\equiv 2\pmod 3$$

then as the least residue system modulo $9$ is

$$\{0,1,2,3,4,5,6,7,8\}$$

we see that $x\equiv 2\pmod 3$ when $x=2,5,8$. Therefore, the solutions are $x\equiv 2,5,8\pmod 9$

Answer 2

Essentially, you divide the modulus by 3 as well, so you get $x\equiv2\pmod 3$, which leads to the solutions $x\equiv2,5,8\pmod 9$.