Mathematics 5336                  Number Theory                          Dr. Cordero

Test #2  Due Monday October 14 at 5:30 p.m.

1. Prove that every set of three consecutive integers contains one which is divisible by 3. Hint: Let the three integers be k, k+1, and k+2. Consider three cases, depending on what remainder you get when you divide k by 3.

1. Suppose m divides a-b and m divides c-d. Prove:
• m  divides (a+c)-(b+d)
• m  divides ac-bd

1. (a). Use the Division Algorithm to find gcd(1721, 378).

(b). Find all solutions to the equation  1721 x + 378 y=14.

1. Find all solutions with x and y positive of the Diophantine equation

6 x + 8 y =120.

1. a. Farmer Jones owes Farmer Brown \$10. Neither has any cash, but Jones has 14 cows, which he values at \$185 each. He suggests paying his debt in cows, with Brown making change by giving Jones some of his pigs at \$110 each. Is this possible, and how?

b. Brown wants his \$10, but says his pigs are worth \$111 each. Is the deal still possible, and how?

c. Jones offers to trade his cows for \$184 each if Brown accepts \$100 per pig. Now what?

6. Reflection:

A part of your grade in this class will be the project. Project presentations will have two parts: a written paper and an oral presentation. For this “reflection” I want you to spend some time thinking about how we should structure the projects so that they will be most useful to you as a (future) teacher. Here are some questions to consider:

·         Should the project be done individually or in groups of two or three members?

·         Should the oral presentations be prepared as if the audience were middle or high school students? Or should we “stick to reality”?

·         Devise a rubric for grading the projects. Imagine 100 points being possible for the written and oral components combined. How much of that should go to the written presentation? How much for the oral? What exactly do you think I should be looking for in each component? How many points (if any) should be allotted for your classmates’ assessment of your oral presentation?

Write a few paragraphs addressing these questions. Please include any other thoughts you have on this issue as well.