Where's my error on finding all the solutions of a linear congruence? I'm supposed to find all solutions of each of the linear congruence.
9x ≡ 5 (mod 25)

I know there are other posts on the site about this, but I don't really follow.

Here's what I did:
I used the Euclidean Algorithm to find the gcd, which was 1 and then to find the equation, I ended up with
1=(4)25 - (11)9
Then I multiplied by 5 on both sides to get it in the form of the original and got:
5=(20)25 - (55)9
Then 55(9) - 25(20) = 5
So I had x ≡ 55 (mod 25) or x ≡ 5 (mod 25).
But the book had x ≡ 20 (mod 25)
What did I do wrong?

Here's my exact work using Euclidean:  My sign seems correct though.


There is a much better way to look at this.
The whole idea of modulos is that $9x=5+25 \cdot n$ for any integer $n$.
What you can do then is keep adding 25 to the right hand side until it is divisible by 9. As it turns out 180 is the first such number so we have 
$$9x \equiv 180 \pmod{25}$$ as gcd(9,25)=1 we can divide by 9 to get
$$x \equiv 20 \pmod{25}$$
You only have to sub your answer back in to see that it's wrong.
