Extended Euclidean Algorithm

We have $9^2\equiv -1 \pmod{41}$ so $9\times (-9)\equiv 9\times \boxed{32} \equiv 1$ .

The solutions of the diophantine equation $9x + 41y = 1$ are applying Euclidean algorihtm $(x,y) = (-9 + 41\lambda, 2 - 9\lambda)$ with $\lambda \in \mathbb{Z} \Rightarrow$ the smallest positive value of $x$ is $-9 + 41 = 32$

Hello By method of Bash or Guess and Check , we find the answer is $\boxed{32}$ (Guess and Check on 41x+1, then check if its divisible by 9, if it is, then SCORE). Thank