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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 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 = ...

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.