Finding all the solutions of a linear equations

Question

I am trying to find all the solutions to the following equation:

$5x \equiv 15\pmod{25}$

Here is what I've done:

1. Find the $\mathrm{gcd}(5,25) = 5$; there will be $5$ solutions.

2. Divide the original equation with the $\mathrm{gcd}$; this turns the equation into $x \equiv 3\pmod{5}$

3. $x ≡ 3 \pmod{15} \to 3 = 15\cdot 1 - 12$

4. $-12 \pmod{15} = 3$.

So the first solution I found would be $(3,3)$ but this is not correct.

If anyone can help with this problem, I'd appreciate it. Thanks!

Answer 1

We have

$$5x\equiv 15\pmod{25}\iff 5x=15+25k,\;k\in\Bbb Z\iff x=3+5k,\; k\in\Bbb Z\iff x\equiv 3\pmod5$$