1 RYERSON UNIVERSITY DEPARTMENT OF MATHEMATICS Midterm Test - MTH 617 Last Name (Print): Signature: First Name: . Section Number: . Student Number: . Date: March 23rd, 2016 3:15 - 6:00 pm Instructions: 1. Enter your section number NOW. 2. Have your student card available on your desk. 3. This is a closed-book test. You are given one page of formulas. Notes, calculators and other aids are not permitted. Verify that your test booklet has pages 2-6. 4. DO NOT ASK FOR ANY INTERPRETATION OF ANY QUESTIONS ON For Instructor's use only. THIS TEST. 5. This test consists of 2 parts. Question(s) (a) Part A has 17 multiple-choice questions. There are no part marks in the multiple-choice section. Each correct response receives 2 points. An 1-17 incorrect response or no response receives 0 point. Write down your responses for each of these questions on page 3. Only the responses 18 entered there will be graded. (b) Part B has 4 full-solution questions with the mark for each question as 19 indicated. i. Unless otherwise instructed, make sure you include all significant 20 steps in your solutions to questions in Part B, presented in the correct order. Unjustified answers will be given little or no credit. Cross out 21 or erase all rough work not relevant to your solution. ii. Present your solutions neatly and legibly in the space provided. If you need more space, use the back of the page. Indicate this fact on the original page, making sure that your solution cannot be confused with any rough work which may be there. iii. The maximum score attainable on this test is 64. Your grade will be computed out of 54. Thus, it is possible to achieve 64/54 on this test. (Too generous, maybe?) Total Score 2 Part A - Multiple Choice Questions. There is only one correct answer for each question. Write down your answers at the bottom of Page 3. Question 1 (23 102 ) = (a) 360 (b) 720 (c) 80 (d) 320 (e) 1 < p < 2 Question 2 [17802 ]100 = (a) 17 (b) 172 (c) 89 (d) 71 (e) 1 < p < 2 Question 3 In the group Z25 , the inverse of 3 is equal to the inverse of (a) 9 (b) 12 (c) 19 (d) 23 (e) 1 < p < 2 Question 4 If [a]12 = 4, [b]12 = 3, then [3a2 2b3 ]12 equals (a) 6 (b) 8 (c) 9 (d) 6 (e) 1 < p < 2 Question 5 Suppose the inverse of a is Z100 is 7 and the inverse of b in Z100 is 29. The inverse of ab in Z100 is (a) 1 (b) 2 (c) 3 (d) 2 (e) none of these Question 6 Let a R be a constant. Define x y = x + ay for all x, y in R. The number of different values of a for which is associative on R is (a) 1 (b) 2 (c) 3 (d) 4 (e) none of these Question 7 Suppose that a is an element in a group and ord(a) = 27, then |ha6 i| = (a) 6 (b) 9 (c) 12 (d) 15 (e) none of these Question 8 Which of the following group is not cyclic: (a) Z (c) Z4 (b) Z20 (d) Z9 (e) none of these Question 9 Suppose that a, b, x are elements in the same group with neutral element e. If ax2 = b and x3 = e, then x = (a) ab1 (b) a1 b (c) b1 a (d) ba (e) none of these Question 10 Suppose that a, x are elements in the same group with x2 = a5 and ord(x) = 6. If ord(a) = p, where p is a prime, then p = (a) 2 (b) 3 (c) 5 (d) 7 (e) none of these Question 11 Suppose that G is a group of size 187. Let H, K be subgroups of G with |H| = 6 |K|, H 6= G, K 6= G. Then |H K| = (a) 1 (b) 17 (c) 11 (d) 187 (e) none of these Question 12 Let G = {(x, y) : x R, y R, y 6= 0}. For all (a, b), (c, d) in G, define (a, b) (c, d) := (ad + bc, bd). Then G is a group under . The neutral element of this group G is (a) (1, 1) (b) (1, 1) (c) (1, 1) (d) (1, 1) (e) none of these 3 Question 13 Let S := {x Z37 : x = x1 }, then |S| = (a) 1 (b) 2 (c) 4 (d) d for some d > 4 (e) none of these Question 14 Suppose that a is an integer greater than 1 and p is a prime and p > 5. If gcd(a, p) = 1, then the number of possible values of gcd(3a, 2a + p) is (a) 1 (b) 2 (c) 3 (d) 4 (e) none of these Question 15 Let a = 1432016 + 2016143 . Then [a60 ]143 = (a) 14 (b) 16 (c) 3 (d) 1 (e) none of these (d) 13 (e) none of these Question 16 [26!]29 [2727 ]29 = (a) 2 (b) 0 (c) 3 Question 17 The number of integers x with 1000 x 2000 that satisfies the equation 5x 18 (mod 22) is (a) 43 (b) 44 (c) 45 (d) 46 (e) none of these (Hint: x0 = 8, y0 = 1 satisfies the equation.) Using on erasable ink, write down your answers using A, B, C, D or E for the seventeen MC questions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 4 Part B - Full solution problems Question 18 (4+6) 1. Suppose that a, b, c are integers with gcd(a, b) = 1 and a|bc. Show that a|c. 2. Let a Z with gcd(a, 1764) = 1. Show that [a42 ]1764 = 1. (Hint: 1764 = 4 9 49. Apply Euler's theorem three times.) 5 Question 19 (8) Let G be a group and let H < G. Let a, b be elements of G. If Ha Hb 6= , show that Ha = Hb. (Recall that for any g G, we defined Hg := {hg : h H}.) 6 Question 20 (6) Let G be a group of size 4. Show that either G is cyclic or every element of G is its own inverse. Question 21 (6) If 1 d 7 and d is an integer, then there is no group which has exactly d elements of order 20. True or false? If it is true, prove it. If it is false, give an example of a group G and an integer d, with 1 d 7, for which G has exactly d elements of order 20