Number Theory Review 2

    Here is a list of 35 problems that will help a number theory student understanding their working knowledge and skill level for elementary number theory.

(1) Find the prime factorization of

(2) Show that if and are positive integers and then

(3) Show that if and then

(4) Find the least common multiple of each of the following pairs of integers:

(5) Show that if and are integers, then if and only if and

(6) Find the prime factorization of

(7) Find the prime factorization of using the Fermat factorization method.

(8) Find the prime factorization of using the Fermat factorization method.

(9) Find the prime factorization of using the Euler factorization method.

(10) Show that any number of the form can be factored easily and then show how to factor

(11) Solve the linear Diophantine equation by either finding all solutions or by showing there are none for

(12) Solve the linear Diophantine equation by either finding all solutions or by showing there are none for

(13) A grocer order apples and oranges at a total cost of If apples cost him each and oranges cost him each, how many of each type of fruit did he order?

(14) A postal clerk has only and stamps to sell. What combination of these may be used to mail a package requiring postage of exactly

(15) Solve the linear Diophantine equation by either finding all solutions or by showing there are none for

(16) Show that each of the following congruences hold

(17) Determine whether each of the following pairs of integers is congruent modulo7.

(18) Show that if and are integers such that and then

(19) What can you conclude if where and are integers and is prime?

(20) Show by mathematical induction that if is a positive integer,  then

(21) Find all solutions to the linear congruence

(22) Find all solutions to the linear congruence

(23) Find all solutions to the linear congruence

(24) Determine which integers where have an inverse modulo 14 and for each one find its inverse.

(25) Show that if is an inverse of modulo and is an inverse of modulo then is an inverse of modulo

(26) Find all the solutions of the system of linear congruence equations:

(27)  Find all the solutions of the system of linear congruence equations:

(28)  Find all the solutions of the system of linear congruence equations:

(29) If eggs are removed form a basket 2, 3, 4, 5, and 6, at a time, there remain, respectively, 1, 2, 3, 4, and 5 eggs. But if the eggs are removed 7 at a time, no eggs remain. What is the least number of eggs that could have been in the basket?

(30)  Find all the solutions of the system of linear congruence equations:

(31) Find all solutions of the polynomial congruence equation:

(32) Find all solutions of the polynomial congruence equation:

(33) Find all solutions of the polynomial congruence equation:

(34) Find all solutions of the polynomial congruence equation:

(35) How many incongruent solutions are there to the polynomial congruence equation

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