TITLE: Solving $10x \equiv 1 \pmod{11^2}$
QUESTION [5 upvotes]: $$10x \equiv 1 \pmod{11^2}$$
I know $10x \equiv 1 \pmod{11}$.
I Have $x \equiv 109 \pmod{11^2}$ but this took me a lot of time to find. 
Is there a quicker way?

REPLY [4 votes]: The systematic way is to use the extended Euclidean algorithm for $\gcd(10,11^2)$ and get the inverse of $10 \bmod 11^2$, which is $109$.