4. For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) If a E Z is even, then gcd(a, a + 2) = 2. (b) If a and b are nonzero integers and d is common divisor of a and b, then d E aZ+ bZ.
4. For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) If a E Z is even, then gcd(a, a + 2) = 2. (b) If a and b are nonzero integers and d is common divisor of a and b, then d E aZ+ bZ.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.1: Definition Of A Ring
Problem 32E: 32. Consider the set .
a. Construct addition and multiplication tables for, using the...
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![4. For each of the following statements, indicate whether the statement is true or false and
justify your answer with a proof or a counterexample.
(a) If a E Z is even, then gcd(a, a + 2) = 2.
(b) If a and b are nonzero integers and d is common divisor of a and b, then de aZ+ bZ.
(c) Elements x and y in a ring R are relatively prime if 1R is a greatest common divisor of
x and y. If a, BE Z[i] and ged(N(@), N(B)) = 1, then a and B are relatively prime.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26930520-9d20-4f6c-8407-b9f9be1c91f3%2Fffc21877-28fd-4a7a-b0c0-ec6afaa013c7%2Ffn56giq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. For each of the following statements, indicate whether the statement is true or false and
justify your answer with a proof or a counterexample.
(a) If a E Z is even, then gcd(a, a + 2) = 2.
(b) If a and b are nonzero integers and d is common divisor of a and b, then de aZ+ bZ.
(c) Elements x and y in a ring R are relatively prime if 1R is a greatest common divisor of
x and y. If a, BE Z[i] and ged(N(@), N(B)) = 1, then a and B are relatively prime.
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