# 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

and

gcd(p,q) = 1

and

let a and b be integers with

0<=a <=p-1

and

0<=b<=p-1

4) Prove that there exists an integer x such that

(x mod p) mod q = a

and

(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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