   Chapter 8.3, Problem 17ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove that for all integers m and n,m=n (mod 3) if, and only if, m   mod   3 = n   mod 3. Prove that for all integers m and n any positive integer d , m = n ( mod   d ) if, and only if, m   mod   d = n   mod   d .

To determine

(a)

To prove:

For all integers m and n, m = n ( mod 3 ) if, and only if, m mod 3 = n mod 3.

Explanation

Given information:

m and n, are integers.

Proof:

Let m and n be integers.

Necessary condition:

Let us assume mn (mod 3). By the definition of m being congruent to n modulo 3:

3|(mn)

By the definition of divides, there exists an integer k such that:

mn=3k

m=n+3k

Take modulo 3 of each side of the previous equation:

m mod 3=(n+3k) mod 3

We can simplify the right side, since (3k)

To determine

(b)

To prove:

For all integers m and n and any positive integer d, m = n (mod d ) if, and only if, m mod d = n mod d.

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