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Practice Exam 3

Problem (1) Using Mathematical Induction prove,

Problem (2) Using Mathematical Induction prove,

Problem (3) Show that if is an integer, then divides

Problem (4) Show that the fourth power of every odd integer is of the form

Problem (5) Show that if is a prime number, then is composite.

Problem (6) Show that if is a perfect square and is prime, then

Problem (7) Show that and are relatively prime for all integers .

Problem (8) Show that if and are relatively prime integers, then 1 or 3.

Problem (9) Apply the Euclidean Algorithm to find

Problem (10) Apply the Euclidean Algorithm to find

Problem (11) Find the unique factorization of

Problem (12) Find the unique factorization of

Problem (13) Determine how many solutions to

Problem (14) Find all solutions to

Problem (15) Show that every integer of the form with is composite.

Problem (16) Apply the Euclidean Algorithm to solve Then find all solutions.

Problem (17) Show that if is an even integer and is an odd integer, then

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