Find all solutions to $2x \equiv p \mod 3p$

Question

Find all solutions to $2x \equiv p \pmod {3p}$. $p$ is prime, and $p > 3$.

I found that this is equal to $2x = p(3k+ 1)$ for some $k \in \Bbb{N}$. Since $k$ can't be even, then we have $2x = \{4p, 10p, 16p, 22p\}$ so the solution set is $x = \{(2+3s)p\}$, for some $s \in \Bbb{N}$.

Is this a valid solution?

Answer 1

Yes, your solution is perfectly right, i.e., $x = p(3s+2)$, where $s \in \mathbb{Z}$.

Answer 2

If $\,p\,$ is any odd integer then $\,{\rm mod}\ 3p\!:\,\ 2x\equiv p\equiv -2p\iff x\equiv -p\equiv 2p$