Find all integer solutions to linear congruences:
\begin{align} &(a) &3x &\equiv 24 \pmod{6},\\ &(b) &10x &\equiv 18 \pmod{25},\\ \end{align}
What I have so far:
$$(a) \gcd(3,6)=3$$ And we know $3|24$ so there are $3$ solutions. By inspection we know that $x=8$ is a solution.
One, question I have is, even though $x=8$ is a solution, I can also see that $x=2$ is a solution, and among others. Does it matter which I choose, or do I randomly choose a solution I see? From here I'm a tad confused. I'm missing something very simple.
$$(b) \gcd (10,25)=5$$
And $5 \nmid 18$ so there are no solutions. Right?