1. Equations have either no solution or exactly three incongruent solutions.
A. TRUE.
B. FALSE.
C. Nothing can be said.
D. None of the mentioned.
Answer= FALSE
2. Find the solution of x2? 3 mod 11.
A. x ? -9 mod 11 and x? 9 mod 11.
B. x ? 9 mod 11.
C. No Solution.
D. x ? 5 mod 11 and x ? 6 mod 11.
Answer= x ? 5 mod 11 and x ? 6 mod 11
3. Find the solution of x2? 2 mod 11.
A. No Solution.
B. x ? 9 mod 11.
C. x ? 4 mod 11.
D. x ? 4 mod 11 and x ? 7 mod 11.
Answer= No Solution
4. Find the set of quadratic residues in the set –Z11* = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
A. QR set = {1, 2, 4, 5, 9} of Z11*.
B. QR set = {1, 3, 6, 5, 9} of Z11*.
C. QR set = {1, 3, 4, 9,10} of Z11*.
D. QR set = {1, 3, 4, 5, 9} of Z11*.
Answer= QR set = {1, 3, 4, 5, 9} of Z11*
5. In Zp* with (p-1) elements exactly: (p – 1)/2 elements are QR and (p – 1)/2 elements are QNR..
A. TRUE.
B. FALSE.
C. Nothing can be said.
D. None of the mentioned.
Answer= TRUE
6. Find the set of quadratic residues in the set – Z13* = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,12}.
A. QR { 1, 2, 4,5, 10, 12}.
B. QR { 2, 4, 5, 9, 11, 12}.
C. QR { 1, 2, 4,5,10, 11}.
D. QR { 1, 3, 4, 9, 10, 12}.
Answer= QR { 1, 3, 4, 9, 10, 12}
7. Euler's Criterion can find the solution to x2 ? a (mod n)..
A. TRUE.
B. FALSE.
C. Nothing can be said.
D. None of the mentioned.
Answer= FALSE
8. Find the solution of x2? 15 mod 23 has a solution..
A. TRUE.
B. FALSE.
C. Nothing can be said.
D. None of the mentioned.
Answer= FALSE
9. Find the solution of x2? 16 mod 23.
A. x = 6 and 17.
B. x = 4 and 19.
C. x = 11 and 12.
D. x = 7 and 16.
Answer= x = 4 and 19
10. Find the solution of x^2?3 mod 23.
A. x?±16 mod 23.
B. x?±13 mod 23.
C. x?±22 mod 23.
D. x?±7 mod 23.
Answer= x?±16 mod 23
11. Find the solution of x2? 2 mod 11 has a solution..
A. TRUE.
B. FALSE.
C. Nothing can be said.
D. None of the mentioned.
Answer= FALSE
12. Find the solution of x2?7 mod 19.
A. x?±16 mod 23.
B. x?±11 mod 23.
C. x?±14 mod 23.
D. x?±7 mod 23.
Answer= x?±11 mod 23
13. If we use exponentiation to encrypt/decrypt, the adversary can use logarithm to attack and this method is very efficient..
A. TRUE.
B. FALSE.
C. Nothing can be said.
D. None of the mentioned.
Answer= FALSE
14. ?(231)=
A. 230.
B. 60.
C. 80.
D. 120.
Answer= 120
15. n is prime if and only if n divides (2n – 2)..
A. TRUE.
B. FALSE.
C. Nothing can be said.
D. None of the mentioned.
Answer= FALSE
16. Find x for the CRT when x= 2 mod 3; x= 3 mod 5; x = 2 mod 7.
A. 33.
B. 22.
C. 23.
D. 31.
Answer= 23
17. Consider a function: f(n) = number of elements in the set {a: 0 <= a < n and gcd(a,n) = 1}. What is this function?.
A. Primitive.
B. Totient.
C. Primality.
D. All of the mentioned.
Answer= Totient
18. The inverse of 49 mod 37 is
A. 31.
B. 23.
C. 22.
D. 34.
Answer= 34
19. Six teachers begin courses on Monday Tuesday Wednesday Thursday Friday and Saturday, respectively, and announce their intentions of lecturing at intervals of 2,3,4,1,6 and 5 days respectively. Sunday lectures are forbidden. When first will all the teachers feel compelled to omit a lecture? Use CRT..
A. 354.
B. 371.
C. 432.
D. 213.
Answer= 371
20. How many primitive roots are there for 25?.
A. 4.
B. 5.
C. 7.
D. 8.
Answer= 8
21. 17 x2 = 10 ( mod 29 ).
A. x = 3, 22 (mod 29).
B. x = 7, 28 (mod 29).
C. x = 2, 27 (mod 29).
D. x = 4, 28 (mod 29).
Answer= x = 2, 27 (mod 29)
22. x – 4x – 16 = 0 (mod 29).
A. x = 6, 24 (mod 29).
B. x = 9, 24 (mod 29).
C. x = 9, 22 (mod 29).
D. x = 6, 22 (mod 29).
Answer= x = 9, 24 (mod 29)
23. x7 = 17 (mod 29).
A. x = 8, 9, 12, 13, 15, 24, 28 (mod 29).
B. x = 8, 10, 12, 15, 18, 26, 27 (mod 29).
C. x = 8, 10, 12, 15, 17, 24, 27 (mod 29).
D. x = 8, 9, 13, 15, 17, 24, 28 (mod 29).
Answer= x = 8, 10, 12, 15, 18, 26, 27 (mod 29)
24. The inverse of 37 mod 49 is
A. 23.
B. 12.
C. 4.
D. 6.
Answer= 4
25. How many primitive roots are there for 19?.
A. 4.
B. 5.
C. 3.
D. 6.
Answer= 6
26. Find the order of the group G = <Z12*, ×>?.
A. 4.
B. 5.
C. 6.
D. 2.
Answer= 4
27. Find the order of the group G = <Z21*, ×>?.
A. 12.
B. 8.
C. 13.
D. 11.
Answer= 12
28. Find the order of group G= <Z20*, x>.
A. 6.
B. 9.
C. 10.
D. 8.
Answer= 8
29. Find the order of group G= <Z7*, x>.
A. 6.
B. 4.
C. 3.
D. 5.
Answer= 6
30. In the group G = <Zn*, ×>, when the order of an element is the same as order of the group (i.e. f(n)), that element is called the Non – primitive root of the group..
A. TRUE.
B. FALSE.
C. Nothing can be said.
D. None of the mentioned.
Answer= FALSE
31. In the order of group G= <Z20*, x>, what is the order of element 17?.
A. 16.
B. 4.
C. 11.
D. 6.
Answer= 4
32. The order of group G= <Z9, x> , primitive roots of the group are
A. 8 , Primitive roots- 2,3.
B. 6 , Primitive roots- 5.
C. 6 , Primitive roots- 2,5.
D. 6 , Primitive roots- 5,7.
Answer= 6 , Primitive roots- 2,5
33. Which among the following values: 17, 20, 38, and 50, does not have primitive roots in the group G = <Zn*, ×>?.
A. 17.
B. 20.
C. 38.
D. 50.
Answer= 20
34. Find the number of primitive roots of G=<Z11*, x>?.
A. 5.
B. 6.
C. 4.
D. 10.
Answer= 4
35. Find the primitive roots of G=<Z11*, x>?..
A. {2, 6, 8}.
B. {2, 5, 8}.
C. {3, 4, 7, 8}.
D. {2, 6, 7, 8}.
Answer= {2, 6, 7, 8}
36. Find the primitive roots of G = <Z10*, ×>..
A. {2, 6, 8}.
B. {3,6 ,9}.
C. {3, 7, 8}.
D. {3, 7}.
Answer= {3, 7, 8}
37. gcd( 18,300) =
A. 4.
B. 12.
C. 8.
D. 6.
Answer= 6
38. ?(37)=
A. 24.
B. 22.
C. 13.
D. 36.
Answer= 36
39. ?(35)=
A. 24.
B. 25.
C. 22.
D. 18.
Answer= 24
40. ?(21)=
A. 10.
B. 12.
C. 8.
D. 14.
Answer= 12
41. 73 mod 19 =
A. 18.
B. 1.
C. 14.
D. 12.
Answer= 1
42. 7(3+j) mod 19 =
A. 7j mod 19.
B. 1 mod 19.
C. 73 + 7j mod 19.
D. All of the mentioned are true.
Answer= 7j mod 19
43. What is the period of 7m mod 19?.
A. 2.
B. 3.
C. 4.
D. 5.
Answer= 3
44. ?(19)=
A. 14.
B. 13.
C. 18.
D. 17.
Answer= 18
45. What is the period of 11 (mod 19).
A. 2.
B. 3.
C. 4.
D. 5.
Answer= 3
46. What is the period of 17 (mod 19).
A. 5.
B. 7.
C. 9.
D. 11.
Answer= 9
47. What is the period of 9 (mod 19).
A. 12.
B. 10.
C. 11.
D. 9.
Answer= 9
48. How many primitive roots does Z<19> have?.
A. 5.
B. 8.
C. 7.
D. 6.
Answer= 6
49. What is the Discrete logarithm to the base 10 (mod 19) for a =7?.
A. 12.
B. 14.
C. 8.
D. 11.
Answer= 12
50. 3201 mod 11 =
A. 3.
B. 5.
C. 6.
D. 10.
Answer= 3
51. Find a number x between 0 and 28 with x^85 congruent to 6 mod 29..
A. 22.
B. 12.
C. 6.
D. 18.
Answer= 6
52. What is the Discrete logarithm to the base 13 (mod 19) for a =13?.
A. 14.
B. 1.
C. 8.
D. 17.
Answer= 1
53. What is the Discrete logarithm to the base 15 (mod 19) for a =9?.
A. 3.
B. 7.
C. 12.
D. 4.
Answer= 4
54. Find a number x between 0 and 28 with x85 congruent to 6 mod 35..
A. 6.
B. 32.
C. 8.
D. 28.
Answer= 6
55. Find a number 'a' between 0 and 72 with 'a' congruent to 9794 mod 73..
A. 53.
B. 29.
C. 12.
D. 37.
Answer= 12
56. What is the Discrete logarithm to the base 2 (mod 19) for a =7?.
A. 3.
B. 4.
C. 6.
D. 9.
Answer= 6
57. ?(41)=
A. 40.
B. 20.
C. 18.
D. 22.
Answer= 40
58. ?(27)=
A. 6.
B. 12.
C. 26.
D. 18.
Answer= 18
59. Find a number 'a' between 0 and 9 such that 'a' is congruent to 7^1000 mod 10..
A. 2.
B. 1.
C. 3.
D. 4.
Answer= 1
60. ?(440)=
A. 200.
B. 180.
C. 160.
D. 220.
Answer= 160
61. GCD(n,n+1) = 1 always..
A. TRUE.
B. FALSE.
C. Nothing can be said.
D. None of the mentioned.
Answer= TRUE