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Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can be false. Justify your answer with a proof or provide a counterexample. (i) If c + d = 0 (mod m), then gcd(c,m) = gcd(d, m). (ii) If gcd(c, m) = gcd(d, m), then c+d=0 (mod m).

Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can be false. Justify your answer with a proof or provide a counterexample. (i) If c + d = 0 (mod m), then gcd(c,m) = gcd(d, m). (ii) If gcd(c, m) = gcd(d, m), then c+d=0 (mod m).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.5: Congruence Of Integers
Problem 4TFE: Label each of the following statements as either true or false. a is congruent to b modulo n if and...
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Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can be false. Justify your answer with a proof or provide a counterexample. (1) If c + d= 0 (mod m), then gcd(c, m) = gcd(d, m). (ii) If gcd(c, m) = gcd(d, m), then c+d =0 (mod m).
Transcribed Image Text:Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can be false. Justify your answer with a proof or provide a counterexample. (1) If c + d= 0 (mod m), then gcd(c, m) = gcd(d, m). (ii) If gcd(c, m) = gcd(d, m), then c+d =0 (mod m).
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