1. The problem statement, all variables and given/known data Show that given x mod p = y mod p and x mod q = y mod q, the following is true: x mod pq = y mod pq. p and q are distinct primes. 3. The attempt at a solution Here is the proof from someone that I am trying to understand: In general, x≡y (mod p) and x≡y (mod q) ⇒ x≡y (mod LCM(p,q)). Proof. x≡y (mod p) and x≡y (mod q) implies p|x-y and q|x-y implies LCM(p,q)|x-y, which means x≡y (mod LCM(p,q)). (Q.E.D.) So, if p and q are different primes, x≡y (mod p) and x≡y (mod q) yield x≡y (mod pq). I do not understand the proof. Primarily what does p|x-y mean? or any notation with |. Also, how does the LCM(p,q) come in to this?