Originally Posted by
ThePerfectHacker You have: $\displaystyle x\equiv 3(\bmod 7),x\equiv 4(\bmod 11),x\equiv 8(\bmod 13)$. The congruences can be written as $\displaystyle x\equiv 3 - 7\cdot 3(\bmod 7)$, $\displaystyle x\equiv 4 - 2\cdot 11 (\bmod 11)$. This gives $\displaystyle x\equiv -18(\bmod 7)$ and $\displaystyle x\equiv -18(\bmod 11)$, this combines into $\displaystyle x\equiv -18(\bmod 77)$. We can write, $\displaystyle x\equiv - 18 - (77)(26) (\bmod 77)$ and $\displaystyle x\equiv 8 - (13)(156)(\bmod 13)$. Therefore, $\displaystyle x\equiv -2020 \equiv 983(\bmod 1001)$