UNIVERSITY OF MANITOBA DATE: December 7, FINAL EXAMINATION TITLE PAGE TIME: 3 hours EXAMINER: M. Davidson
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1 TITLE PAGE FAMILY NAME: (Print in ink) GIVEN NAME(S): (Print in ink) STUDENT NUMBER: SEAT NUMBER: SIGNATURE: (in ink) (I understand that cheating is a serious offense) INSTRUCTIONS TO STUDENTS: This is a 3 hour exam. Please show your work clearly. A single line display, simple calculator is permitted. No texts, notes, or other aids are permitted. There are no cellphones or electronic translators, or other electronic devices permitted. This exam has a title page and 13 pages of questions, which includes one page with a table of primes. Please check that you have all the pages. You may remove the table if you wish, but be careful not to loosen the staples. The value of each question is indicated in the lefthand margin beside the statement of the question. The total value of all questions is 110 points. Answer all questions on the exam paper in the space provided beneath the question. If you need more room, you may continue your work on the reverse side of the page, but CLEARLY INDICATE that your work is continued. Question Points Score Total: 110
2 PAGE: 1 of 13 [10] 1. (a) Find (2002, 897). (b) Find all integer solutions to 2002 x y = (2002, 897).
3 PAGE: 2 of 13 [10] 2. For each of the following linear congruences, find out how many solutions there are. If solutions exist, you need NOT find them, but you should state a reason for your answer. (a) 3388 x 42 (mod 7413) (b) 2500 x 42 (mod 2012)
4 PAGE: 3 of [8] Given the public information of an RSA encryption is (n, e) = (2599, 107), find the decrypt key d. [Hint: One of the two prime factors of n is less than 30.]
5 PAGE: 4 of 13 [12] 4. Recall d(n) is the number of divisors of n, σ(n) is the sum of the divisors of n and φ(n) is the Euler phi function. ( = ) (a) What is d(229320)? σ(229320)? φ(229320)? (b) Show that if n is a square then d(n) is odd. (c) Under what conditions is φ(2n) = φ(n)? (Justify your answer.)
6 PAGE: 5 of 13 [8] 5. (a) Define what is meant for a number n to be abundant. (b) Define what is meant for a number n to be deficient. (c) For what values of a is 2 a 11 abundant? (d) Show that there are infinitely many deficient numbers.
7 PAGE: 6 of 13 [10] 6. (a) Use Wilson s Theorem to find the least residue of 235! (mod 239). (b) Use Gauss s Lemma to decide if 3 is a quadratic residue or quadratic nonresidue modulo 31. (No credit will be given for any other method.)
8 PAGE: 7 of 13 [18] 7. (a) How many primitive roots does the prime 71 have? (b) What are the possible orders a modulo 71 when (a, 71) = 1? (c) Show that 7 is a primitive root of 71. Continued on next page.
9 PAGE: 8 of 13 (d) List two other primitive roots. (How do you know they are primitive roots?) These should be in least residue. (e) Given that 7 6 = = 71(1657) + 2, what is the order of 2 modulo 71? What is the order of 14 mod 71?
10 PAGE: 9 of 13 [6] 8. Suppose that a has order t (mod m). What is the order of a 2 if: (a) t is odd? (Justify your answer.) (b) t is even? (Justify your answer.)
11 PAGE: 10 of 13 [12] 9. Calculate the following Legendre Symbols: Note: 1723 is prime. ( ) 499 (a) 1723 (b) ( )
12 PAGE: 11 of 13 [8] 10. We note that for the prime number p = 2819, that 2p + 1 = 5639 is also prime. Use Euler s Criterion for quadratic residues (together with calculation of Legendre symbols) to decide which, if any, of 2, 3, 5, or 7 are primitive roots of 5639.
13 PAGE: 12 of 13 [8] 11. (a) Give a formula for finding integer solutions to x 2 + y 2 = z 2. (b) Under what conditions is this solution fundamental? (c) Find a Pythagorean triple where one of the values is: i. 11. ii. 14.
14 PAGE: 13 of 13 NAME: (Print in ink) Fill in the above if you wish to remove this sheet from the exam paper The following is a list of all primes less than
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