Proving x ≡ y (mod pq) from congruences mod p and q

[SpiffyEh]

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 into this?

 

[susskind_leon]

a|b means a devides b, which means that there exists an integer number n so that b = a*n

 

[SpiffyEh]

Oh ok. That makes more sense. I still don't understand how the LCM comes in but it's starting to come together some more. Thanks!

 

[susskind_leon]

You're welcome. It's weird actually they brought the LCM up, because if p and q are primes, then LCM(p,q)=p*q
Hope that helps

 

[SpiffyEh]

Is there another way to prove this without LCM?

 

[susskind_leon]

don't worry about the LCM. If p and q are primes, then LCM(p,q) = p*q, so it's the same thing!