Explanation Given information: m and n, are integers. Proof: Let m and n be integers. Necessary condition: Let us assume m ≡ n (mod 3). By the definition of m being congruent to n modulo 3: 3 | ( m − n ) By the definition of divides, there exists an integer k such that: m − n = 3 k Add n to each side: m = n + 3 k Take modulo 3 of each side of the previous equation: m mod 3 = ( n + 3 k ) mod 3 We can simplify the right side, since ( 3 k ) 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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ISBN: 9780357097717

Author: EPP

Publisher: CENGAGE L

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Chapter 8.3, Problem 17ES

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.

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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Chapter 8.3, Problem 16ES

Chapter 8.3, Problem 18ES

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(a) To prove: For all integers m and n, m = n ( mod 3 ) if, and only if, m mod 3 = n mod 3.

Solution Summary: The author explains that if m and n are integers, they can be proved.