Using Euclidean Algorithm to solve the congruence

Question

Using the Euclidean Algorithm show that $ \gcd(591,607) = 1$

Now find integers $s,t $ such that $591s + 607t = 1 $ and use this to find the value of $x$ that satisfies the congruence $591x= 90\pmod{607} $

I have found s and t to be 37 and -38. However, I am stuck on the last part which is finding $x$.

Here is how I found s and t :

The Euclidean Algorithm gives: 607 = 591* 1 + 16

591 = 36*16 + 15

16 = 15 * 1 + 1

15 = 15* 1 + 0

Then we have 1 = 1* 16 -1* 15

1 = 1*16 -1(591 - 36*16)

1 = -1* 591 + 37(607-591*1)

1 = 37*607 - 591* 38

Therefore $37*607 -38*591 = 1\pmod{607}$

Answer 1

the equation $$591s+607t=1$$ is called a Diophantine equation and the solution is given by $$s=569+607k,t=-554-591k$$ where $k$ is an integer number