I made this calculator to solve system of congruences using Chinese Remainder Theorem a while ago and it's always seemed to work for problems I tried, but today I noticed that the solution was wrong.
(I can't post images - please see this link )
For $S\equiv1$ (mod 41) and $S\equiv2$ (mod 19),
I consider the system as $S\equiv i$ (mod m) and $S\equiv j$ (mod n),
Then rewrite it as $S\equiv jmy + inx$ (mod m*n), where x and y are found by computing using the Extended Euclidean Algorithm.
So as you see in the link, I finally have $1 = ax + by = 41(-6) + 19(13)$, so then $jmy + inx = 2(41)(13) + 1(19)(-6)$, reducing to 173 (mod 779).
But this answer should be 534. What did I do wrong in my steps?