- #1

SpiffyEh

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## Homework Statement

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?