For this question, recall that a Carmichael number is a composite number n such that an-11 mod n whenever ged(a, n) = 1. Let m = P1P2. pr. for distinct primes pi, such that (p:- 1) divides m - 1 for all i. (p₁ (a) If ged(a,m) = 1 show that api-1 = 1 mod pi, for each i. (b) Using (a), show that am-1 = 1 mod p; for each i. (c) Using (b) and the Chinese Remainder Theorem, show that am-1 = 1 mod m.

For this question, recall that a Carmichael number is a composite number n such that an-11 mod n whenever ged(a, n) = 1. Let m = P1P2. pr. for distinct primes pi, such that (p:- 1) divides m - 1 for all i. (p₁ (a) If ged(a,m) = 1 show that api-1 = 1 mod pi, for each i. (b) Using (a), show that am-1 = 1 mod p; for each i. (c) Using (b) and the Chinese Remainder Theorem, show that am-1 = 1 mod m.

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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