1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not divisible by q. Show that the congruence * = a (mod pq) has a unique solution modulo pq whenever the integer...

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Extracted text: 1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not divisible by q. Show that the congruence * = a (mod pq) has a unique solution modulo pq whenever the integer k is relatively prime to (p- 1)(q – 1). 2. Give an example illustrating the result above. 3. Generalize the result in Problem 1 to the case of the congruence * =a (mod pipaPs-Pk) where p1, P2, . Pk are distinct primes. Prove your generalized result. 4. Use the result in Problem 3 and state conditions on n, a, and k under which the congruence * =a (mod n) is solvable.