Discrete mathematics proofs

1) Prove that all integers a,b,p, with p>0 and q>0 that
((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q
Or give a counterexample

2) prove for all integers a,b,p,q with p>0 and q>0 that
((a-b)mod p) mod q=0              
if and only if
(a mod p) mod q = (b mod p) mod q
Or give a counterexample.

3) let p and q be positive integers with
0 < p < q
and
gcd(p,q) = 1
and
let a and b be integers with
0<=a <=p-1
and
0<=b<=p-1

Prove that there exists an integer x such that
(x mod p) mod q = a
and
(x mod q) mod p = b