Discrete Mathematics II Problem 2 Prove the following theore

Discrete Mathematics II

Problem 2: Prove the following theorem or find a counterexample to disprove it.

Theorem: Suppose m, n, d Z. If m mod d = n mod d then m = n.

Solution

We are given that if

m mod d = n mod d

let\'s assume

a= m mod d

we can write it as

m=dk+a

or

a=m-dk ....where k is any integer

now, assume

n mod d =b

n=b+dk

b=n-dk.......where \'k\' can be any number

now, we are given

m mod d = n mod d

or a=b

m-dk = n-dk

dk gets cancelled

and we will get

m=n............Answer