Advanced math problem

Please help to find a deductive technique for this problem. (See attached file for full problem description)

Proof problem

I need help in determining how to deduce a solution to prove the assertion. I must follow directions exactly as asked in the question. (See attached file for full problem description)

Advanced math proof on functions

I need help in proving these kinds of function proof. (See attached file for full problem description)

Number Theory. 400 level. Introductory Course in Undergraduate.

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)

Number Theory. 400 level. Introductory Course in Undergraduate.

Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields. (See attached file for full problem description)