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1 1/ 19 2/17 3/23 4/23 5/18 Total/100 Please do not write in the spaces above. Directions: You have 50 minutes in which to complete this exam. Please make sure that you read through this entire exam before attempting any problems. You must show all work, or risk losing credit. Be sure to answer all questions asked. To receive full credit on problems, they must not only be mathematically correct, but they must also be solved using the correct notation and terminology. The following list of all primes below 100 may or may not be helpful: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 Good luck! MATH Version II Fall 2015 Dr. Morton Name: Exam II
2 1. (19 points) The following list of all primes below 100 may or may not be helpful: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 a. Is 1219 prime? How do you know? Use and completely describe the best method to prove your point (i.e. show all work). b. If we have the number = , how many factors are there? Use the formula and show all work. Do not attempt to find all the factors. (Note: 101 is prime.) c. lllustrate Goldbach s conjecture for the number 82. d. Are the numbers 225 and 60 relatively prime? How do you know? e. Is it possible that 61 is a Mersenne prime? Note that 61 is in the list of primes above. Explain carefully. f. In the proof that 2 is irrational, we start by supposing that 2 is rational. Then we can write 2 = a b where a and b are whole numbers, a and b are relatively prime, and b is non-zero. We can simplify the equation 2 = a in two different steps to make this equation more user friendly, b showing all work. (Then stop here we will only prove this part.)
3 2. (17 points) a. Precisely identify the following figure: b. Draw two different unfoldings of a 3-cube. c. How many different unfoldings of a 3-cube are there? d. How does one make a projection of a 3-dimensional cube? (Answer precisely) e. Carefully and specifically identify the following picture: g. On the following bigger pictures of the figure in part (f) carefully circle and number all vertices on the left hand picture, label and number all the 1-cubes in the middle picture, and label and number all the 2- cubes in the right hand picture. Make sure that it is clear what each of your labels refers to. Vertices 1-cubes 2-cubes h. Precisely identify the following picture:
4 3. (23 points) The following list of all primes below 100 may or may not be helpful: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 a. How many Mersenne primes are known as of today? b. The number is a Mersenne prime (it is the 33 rd Mersenne prime). What is the corresponding perfect number? (Note: This is too large to fit in your calculator; just write what you would enter into your calculator.) c. Is it possible that is also a Mersenne prime? Why or why not? Be precise in your reasoning. d. Use a factor tree to write 266 according to the Fundamental Theorem of Arithmetic. e. How many divisors does 266 have? Use the formula to figure this out, showing all work. f. Find all divisors of 266. No work is required here. g. Is 266 abundant, deficient, or perfect? Show all work (including equalities/inequalities) and give your reasoning and conclusions. h. Is it possible that 266 and 214 are friendly? 214 s divisors are 1,2,107, and 214 (you do not have to check this; it is true). If yes, show why. If no, explain which condition(s) are missing.
5 4. (23 points) a. Completely fill in the open entries of the table of m-dimensional faces of n-cubes: (Note: I have filled in some entries for you. Also some rows and some columns have been skipped.) Any entries that are shaded black may be ignored. Note: some entries are too big to fit into your calculator just tell me what you would have entered. n - c u b e s m- faces b. Suppose you have a 67-dimensional cube (the sides of which are all one unit long). How would you move it to build a 68-dimensional cube? Be specific about your answer. c. Suppose a square moves through Lineland by hitting on one of its vertices. What do the Linelanders see as time passes (i.e. what do they perceive is happening?) Describe this carefully in words, making sure to tell me how the size of the perceived figure is changing. d. Give the coordinates of two different vertices of a 6-dimensional cube.
6 5. (18 points) Prove the following theorem: There are infinitely many primes. I will lead you through the proof and you will supply the missing steps. Note: If you would like to just write your own proof on the reverse side of this page please do so and note that you are doing so. We will be using a proof by contradiction here. a. There are two contradictory statements that could be true concerning this theorem. They are both listed below. Circle the statement with which we will begin our proof. There are infinitely many primes. There are finitely many primes. b. If the circled statement in part a is true, then what must be true about the prime numbers (in terms of large prime numbers)? c. Based on part b, we can generate a special list. 1. What is on the list? 2. List everything on the list (defining all variables). 3. What does it mean to not be on the list? d. Using the list in part c, we can construct a special number X. Carefully construct the number X. e. Is it possible that X is in the list from part e? What tells you this? f. Based on your answer to part e, is X prime or composite? Why? g. From part f, what must be the relationship between a number on the list and the number X? h. Does 2 X? How do you know? Does 3 X? How do you know? Does any number on the list divide X? How do you know? i. From part h, were there any divisors of X on the list? j. We have reached our contradiction. Which parts from the above give us our contradiction? (List the letters of the parts). k. Based on part j, what can be said about the statement that you circled in part a above? What does this prove?
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