modular arithmetic - Finding Multiplicative Inverse

I'm told to find the multiplicative inverse of $\mathbf 9\bmod37$.

I can't really use the the Euclidean Algorithm on the equation $\mathbf 9 = Q \cdot 37 + R$ where the LHS is already smaller than the RHS or am I wrong in thinking this way?

Answer

Hint, easy to observe $9\cdot(-4)=-36=1 \pmod {37}$. On your other point, Euclidean algorithm works fine.