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Let a, be Z and n 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 a = b (mod n), then gcd(a, n) = gcd(b, n). (ii) If gcd(a, n) = gcd(b, n), then a = b (mod n).

Let a, be Z and n 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 a = b (mod n), then gcd(a, n) = gcd(b, n). (ii) If gcd(a, n) = gcd(b, n), then a = b (mod n).

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 a, b e Z and n 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 a = b (mod n), then gcd(a, n) = gcd(b, n). (ii) If gcd(a, n) = gcd(b, n), then a = b (mod n).
Transcribed Image Text:Let a, b e Z and n 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 a = b (mod n), then gcd(a, n) = gcd(b, n). (ii) If gcd(a, n) = gcd(b, n), then a = b (mod n).
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