1. Use Euler's Theorem to prove Q265 = a for all a E Z. a (mod 105)
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 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadf70025-15e9-4d93-968e-9d86049b61bd%2Fb0cf6238-7d85-449f-98a3-ee7b7ac89ba3%2Fy0khx7.jpeg&w=3840&q=75)
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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