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Discrete math proofs

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Please help with the following proofs. Answer true or false for each along with step by step 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
gcd(p,q) = 1
let a and b be integers with
0<=a <=p-1

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

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Solution Summary

There are several discrete math proofs in this solution. All of them involve integers and counter examples. The proofs are provided in step by step format.

Solution Preview

1. False
For example, a=3, b=5, p=7, q=17, then
((a+b) mod p) mod q=1
(a mod p) mod q + (b mod p) mod q =3+5=8
The equation does not hold.

2. False.
For example, a=3, b=-4, p=17, q=7, then
((a-b) mod p) mod q = ...

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