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.
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?