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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

Number Theory Books

A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of N... List Price: $18.95 Buy New: $7.50 You Save: $11.45 (60%) New (32) Used (23) from $7.50The Universe May Be a Mystery,But It's No SecretMichael Schneider leads us on a spectacular, lavishly illustrated journey along the numbers one through ten to explore the mathematical principles made visible (more)

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathem... List Price: $16.00 Buy New: $8.75 You Save: $7.25 (45%) New (38) Used (19) from $7.49In 1859, Bernhard Riemann, a little-known thirty-two year old mathematician, made a hypothesis while presenting a paper to the Berlin Academy titled "On the Number of Prime Numbers Less Than a Given Quantity." (more)

Bayesian Computation with R (Use R) List Price: $49.95 Buy New: $37.86 You Save: $12.09 (24%) New (36) Used (13) from $35.00There has been a dramatic growth in the development and application of Bayesian inferential methods. Some of this growth is due to the availability of powerful simulation-based algorithms to summarize (more)

Euler's Gem: The Polyhedron Formula and the Birth of Topology List Price: $27.95 Buy New: $16.98 You Save: $10.97 (39%) New (31) Used (4) from $16.98Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so (more)

e: The Story of a Number List Price: $19.95 Buy Used: $2.94 You Save: $17.01 (85%) New (11) Used (37) Collectible (1) from $2.94The interest earned on a bank account, the arrangement of seeds in a sunflower, and the shape of the Gateway Arch in St. Louis are all intimately connected with the mysterious number e. In this informal (more)

Not Even Wrong: The Failure of String Theory and the Search for Unity in Phys... List Price: $16.95 Buy New: $7.61 You Save: $9.34 (55%) New (33) Used (11) from $7.27When does physics depart the realm of testable hypothesis and come to resemble theology? Peter Woit argues that string theory isn't just going in the wrong direction, it's not even science. Not Even Wrong (more)

Killer Poker By the Numbers: Mathematical Edge for Winning Play List Price: $14.95 Buy New: $6.99 You Save: $7.96 (53%) New (34) Used (13) from $6.99Killer Poker By the Numbers: Mathematical Edge for Winning Play (more)

Elementary Number Theory (5th Edition) List Price: $124.00 Buy New: $99.20 You Save: $24.80 (20%) New (10) Used (17) from $95.50Elementary Number Theory and Its Applications is noted for its outstanding exercise sets, including basic exercises, exercises designed to help students explore key concepts, and challenging exercises. (more)

Complex Analysis for Mathematics and Engineering List Price: $124.95 Buy New: $44.94 You Save: $80.01 (64%) New (14) Used (14) from $44.94Revised and updated, the new Fifth Edition of Complex Analysis for Mathematics and Engineering presents a comprehensive, student-friendly introduction to Complex Analysis. It's clear, concise writing (more)

Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills List Price: $29.95 Buy New: $14.75 You Save: $15.20 (51%) New (42) Used (16) from $14.49I used to think math was no fun'Cause I couldn't see how it was doneNow Euler's my heroFor I now see why zeroEquals e[pi] i+1--Paul Nahin, electrical engineer In the mid-eighteenth century, Swiss-born (more)