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Proposition 5.5.29. Let n >1 be an integer, and let a be an element of Zn \ {0}. Then a has an inverse in Zn if and only if gcd(a, n)=1 (that is, a is relatively prime to n). The proof of this proposition is up to you:

Proposition 5.5.29. Let n >1 be an integer, and let a be an element of Zn \ {0}. Then a has an inverse in Zn if and only if gcd(a, n)=1 (that is, a is relatively prime to n). The proof of this proposition is up to you:

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.3: The Characteristic Of A Ring
Problem 13E
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Proposition 5.5.29. Let n >1 be an integer, and let a be an element of Zn \ {0}. Then a has an inverse in Zn if and only if gcd(a, n)=1 (that is, a is relatively prime to n). The proof of this proposition is up to you:
Transcribed Image Text:Proposition 5.5.29. Let n >1 be an integer, and let a be an element of Zn \ {0}. Then a has an inverse in Zn if and only if gcd(a, n)=1 (that is, a is relatively prime to n). The proof of this proposition is up to you:
Exercise 5.5.30. Let n > 1 be an integer, and let a be an element of Z, \ {0}. (a) Prove the "only if" part of Proposition 5.5.29. That is, prove that if a has an inverse in Z, \ {0} then gcd(a, n)=1. (*Hint*) 5.5 MODULAR DIVISION 147 (b) Prove the "if" part of Proposition 5.5.29. That is, prove that if gcd(a, n)=1 then a has an inverse in Z, \ {0} . (*Hint*)
Transcribed Image Text:Exercise 5.5.30. Let n > 1 be an integer, and let a be an element of Z, \ {0}. (a) Prove the "only if" part of Proposition 5.5.29. That is, prove that if a has an inverse in Z, \ {0} then gcd(a, n)=1. (*Hint*) 5.5 MODULAR DIVISION 147 (b) Prove the "if" part of Proposition 5.5.29. That is, prove that if gcd(a, n)=1 then a has an inverse in Z, \ {0} . (*Hint*)
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