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1. Use Euler's Theorem to prove Q265 = a for all a E Z. a (mod 105)

1. Use Euler's Theorem to prove Q265 = a for all a E Z. a (mod 105)

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 39E
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1. Use Euler's Theorem to prove a265 = a (mod 105) for all a E Z. 2. Use Fermat's Little Theorem, or its corollary, to find the units digit of 72018 + 112019 + + 132020 + 132021 + 172022 3. Use Wilson's Theorem to prove 6(k – 4)! = 1 (mod k), if k is prime. 4. Use Fermat's factorization method to factor 2168495737. 5. Use Kraitchik's factorization method to factor 11653. 6. Prove o (k2) = k · 4(k) for all k E N. %3D 7. Prove each of the following statements. (a) If q is a prime number not equal to 3 and k = 3q, then o(k) = 2 (T(k) + ¢(k)). %3| %3D (b) If q is an odd prime number and k = 2q, then k = o(k) – T(k) – (k). %3D 8. Let â be the inverse of a modulo k. Prove that the order of a modulo k is equal to the order of â modulo k. Use this result to easily show that if a is a primitive root modulo k then â is also a primitive root modulo k.
Transcribed Image Text:1. Use Euler's Theorem to prove a265 = a (mod 105) for all a E Z. 2. Use Fermat's Little Theorem, or its corollary, to find the units digit of 72018 + 112019 + + 132020 + 132021 + 172022 3. Use Wilson's Theorem to prove 6(k – 4)! = 1 (mod k), if k is prime. 4. Use Fermat's factorization method to factor 2168495737. 5. Use Kraitchik's factorization method to factor 11653. 6. Prove o (k2) = k · 4(k) for all k E N. %3D 7. Prove each of the following statements. (a) If q is a prime number not equal to 3 and k = 3q, then o(k) = 2 (T(k) + ¢(k)). %3| %3D (b) If q is an odd prime number and k = 2q, then k = o(k) – T(k) – (k). %3D 8. Let â be the inverse of a modulo k. Prove that the order of a modulo k is equal to the order of â modulo k. Use this result to easily show that if a is a primitive root modulo k then â is also a primitive root modulo k.
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