How can I determine the original number $x\in[pq]$ from its remainders $x_p$ and $x_q$, when it's divided by two relatively prime numbers $p$ and $q$, given that $\gcd(p, q) = 1$? I learned about a method using Bézout's theorem, which suggests that because $p$ and $q$ are coprime, there must be integers $X$ and $Y$ satisfying $pX + qY = 1$.
The formula to reconstruct $x$ is given as $x = x_pqY + x_qpX$.
Could you explain and provide a proof of why this method accurately reconstructs $x$?