Math 3100/6100
Number Theory Unit Exam
Summer 2004

Directions. Please print a copy of this exam. You may write your solutions on the exam or attach them. Some questions require just a few calculations to demonstrate your solution. Others require a little more explanation. I have tried to be clear on this, but please call or e-mail me if you have any questions.

To give you a time gauge, you should be able to complete this exam in no more than 2 hours (my thought is one hour). However, the exam is take-home so if you need more time, take it. The test is worth 60pts. There are 21 questions worth 3pts each. So, there is a built in “bonus” question… you can miss one and still receive a score of 60/60.

Please bring your completed exam with you to class on Monday (July 19th). Thus, the exam is due no later than 5:45pm on the 19th. If you cannot be at class then you need to make arrangements to have the test to me before this due date and time.

Honor Code. Your signature below communicates to me that you completed this exam entirely on your own. Which means you did not use any of the following resources
1. Any other person. No discussion even with your kids, dog, etc.
2. Any on-line resource (except my webpage, of course)
3. Texts, books, workbooks or any other written resources

This three-step honor code test kit is given to freshman cadets at West Point to help them determine if an action/decision is honorable. I think they are great questions to ask yourself as you work on this exam.
1. Is my action deceiving anyone?
2. Does it gain an advantage I’m not entitled to?
3. Would I be satisfied on the receiving end of it?
 
Name (print) _____________________________________

Signature ________________________________________


 1. Given the following definition of prime numbers, explain why 1 is not a prime number:
     A prime number is a positive integer that has exactly two positive integer factors, 1 and itself.

2. What is the product of the first 5 composite numbers greater than 15?

3. What is the sum of the first 5 prime numbers?

4. Can you express this number (from question 3) as the sum of 4 distinct composite numbers?

5. How could you use skip counting and a sieve (like a hundreds board we used in class) to identify prime #’s

6. A palindrome is a number that is unchanged when its digits are reversed. How many palindromes are there between 1 and 200? How many of these numbers are prime?

(For #7 & 8) Just for fun, let’s say some numbers are more interesting than others. Here’s the criteria we will use to assign an “interesting score” to a number:
Prime Number (15pts)
Perfect Square (10pts)
Perfect Cube (7pts)
Sum of Digits Greater than 14 (4pts)
Even Number (3pts)
Odd Number (2pts)        
Each Factor (1pt)                

7. Pick any two 3-digit numbers and calculate their interest ratings. Which one is more interesting? (Show work)

8. Chose 2 of the criteria above that cannot be applied to the same number. Explain why.

9. Remember the locker game and/or crossing the line outside activity. What mathematical concept did they have in common? (7 letters, begins with F and ends in R S)

Note: The locker problem we worked on in class was posed as follows:
There are 1000 lockers down one side of a long hallway. One hundred students walk down the hallway, one by one. Each student does something different with the lockers. The first student opens all the lockers. The second student closes every other locker. The third student changes the state (open or closed) of every third locker (beginning with the third locker). The forth student changes the state of every forth locker (beginning with the forth locker), and so on. After the 1000th student walks down the hallway, which lockers are open, and which are closed? Why?

10. If you had 20 lockers and 20 students which lockers would be open after the 20th student had changed the 1st and 20th locker’s position?

11. If you had 20 lockers and 20 students but changed the rules slightly so that the 1st students started at the 20th locker, the 2nd student started at the 19th locker, would the result be the same as #10? Why or why not?

12. In the locker problem, which locker number had its position changed (open, close, open, etc…) the greatest number of times 16, 24, or 40?

13. Of these lockers (16, 24, and 40), which students changed the position of all three lockers?

14. (Continued from 13) What is the highest locker number that all three touched? This is called the greatest common factor. What is the greatest common factor of 21, 56, and 63?

15. The Fundamental Theorem of Arithmetic states:
Any positive integer may be expressed as the product of prime numbers (i.e. 20 = 2 x 2 x 5) in one, and only one, way except for the order of the prime factors in the product.
This expression is called the prime factorization of a number. What is the prime factorization of 72?

16. The greatest common factor of 72 and 60 is 12. How could prime factorizations help find the GCF of numbers?

17. Use your knowledge of prime factorization to help you solve the following problem:

The census taker asked Jaclyn about her children. Jaclyn said, "I have 3 daughters, Alice, Betty, and Cindy. The product of their ages is 36. The sum of their ages is the same as the street number of our next door neighbor." The census taker went next door and came back and said: "Still not enough information". Jaclyn said: "Oh, I forgot to tell you that my oldest daughter loves mathematics!" The census taker found out the ages of her daughters immediately. What are their ages?

18. The least common multiple of 8, 5 and 3 is not 90. Why not?

19. We are having a Math Party (not that every day isn’t already a math party!). At the Party Store, paper plates come in packages of 30, paper cups in packages of 15, and napkins in packages of 20. What is the least number of plates, cups and napkins that we need to purchase so that there is an equal number of each?

20. Tom Cruise gets every 4th day off from his schedule while Dr. Sheehy gets every 6th day off from teaching. If Dr. Sheehy met with Tom cruise this afternoon to look over some mathematical investigations, how many days will it be before they can meet again to compare solutions?

21. Is the sum of the month day and year of your birthday a prime or composite number? Since this will be different for most of you, please write down your birthday and sum.