Assignment:
Answer true or false for each along with step by step proofs.
Q1) 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 counter example.
Q2) 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 counter example.
Q3) 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
Q4) Prove that there exists an integer x such that (x mod p) mod q = a and (x mod q) mod p = b
Provide complete and step by step solution for the question and show calculations and use formulas.