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Let p and q be distinct odd primes such that gcd(p−1, q−1) = 2. Does there exist an integer M such that aM ≡ a-1 (mod p) and aM ≡ a (mod q) for all a ∈ (Z/pqZ)*? What if gcd(p−1, q−1) > 2?

Let p and q be distinct odd primes such that gcd(p−1, q−1) = 2. Does there exist an integer M such that aM ≡ a-1 (mod p) and aM ≡ a (mod q) for all a ∈ (Z/pqZ)*? What if gcd(p−1, q−1) > 2?

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 13E
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Let p and q be distinct odd primes such that gcd(p−1, q−1) = 2. Does there exist an integer M such that aM ≡ a-1 (mod p) and aM ≡ a (mod q) for all a ∈ (Z/pqZ)*? What if gcd(p−1, q−1) > 2?

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