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)

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)

3 practice problems

(See attached files for full problem descriptions) I'd like to know the deductive techniques to prove each. A short and correct proof for each is all that is required. OR please provide some help on how to solve them.