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.


$$\text{Hint: Use euclidean algorithm}$$
$$\text{Find $a,b$ such that $9a+25b=1$, which can be done since gcd(9,25)=1}$$
$$\text{You'll finish with: $1 \equiv 9 \cdot a \mod (25) \Rightarrow x \equiv 9x \cdot a (\mod 25)$. Remember $9x \equiv 5 (\mod 25)$}$$
