Modular Arithmetic with 2 Different Moduli

Question

How can I go about solving the following 2 linear congruences?

$x \equiv 2 \pmod 7$

$x \equiv 5 \pmod {11}$

How am I supposed to work with these if they are different moduli?

Any help on how to do this would be greatly appreciated.

Answer 1

Setting $x= 2+7n = 5 + 11m$, you get that $7n - 11m = 3$. In fact, since 7 and 11 are coprime you can do even better - there exist integers $s,t$ such that $7s + 11t = 1$. This is called Bezout's identity and $s$ and $t$ can be found with the extended Euclidean algorithm, but here you can probably just guess and check a solution. Of course once you find $s$ you can triple it to get $n$.

In general, the existence and uniqueness of a solution is given by the Chinese remainder theorem.