If $x$ is a square modulo two primes, then it is a square modulo their product

Question

$a, b$ be integers, $p, q$ primes. If $x \equiv a^2 $ (mod $p$) and $x \equiv b^2$ (mod $q$), then $x \equiv c^2$ (mod $pq$) for some interger $c$.

I attempted to use Chinese Remainer Theorem, but did not get useful forms, so how to prove it?

Answer 1

By the Chinese Remainder Theorem, there exists an element $c$ such that $$c \equiv a \pmod{p} \\ c \equiv b \pmod{q}$$

Then $$x \equiv c^2 \pmod{p}\\ x\equiv c^2 \pmod{q}$$