Given two primes $p$ and $q$, does this following equivalence hold? $$b\equiv a\mod pq\Leftrightarrow\ b\equiv a\mod p\text{ and }b\equiv a\mod q$$
I think "$\Leftarrow$" is true and follows from the CRT although I am not sure how to properly use it. Regarding "$\Rightarrow$", however, I have no idea. It seems to be true although it feels wrong.