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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 37E Chapter 4.1

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 37E

Find counterexamples to each of these statements about congruences.a) If ac ? bc (mod m), where a, b, c, and m are integers with m ? 2, then a ? b (mod m).________________b) If a ? b (mod m) and c ? d (mod m), where a, b, c, d, and m are integers with c and d positive and m ? 2, then ac ? bd (mod m).

Step-by-Step Solution:

Solution:Step 1In this problem we have to find counterexamples to these statements about congruences.we can say integer a is congruent to b (mod) m, if a and b are integers and m is a positive integer and m divides( a-b) completely that is m | (a-b).Or,We can also say a is congruent to b (mod) m. if value of Step 2a) If ac bc (mod m), where a, b, c, and m are integers with m 2, then a b (mod m).For counterexample let, a=4, b=5, c=6, m=6.Here, for ac bc (mod m)We have For ac bc (mod m) to satisfy it should be follow congruence theorem which is true for given equation.now , for a b (mod m).We have, a=4, b=5 and m=6 which does not satisfy congruence theorem that is m | (a-b) for a b (mod m).Hence with counterexample we proved that...

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Chapter 4.1, Problem 37E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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