# Basic concept in Number Theory and Finite fields MCQs

1. Calculate the GCD of 1160718174 and 316258250 using Euclidean algorithm..

A. 882.

B. 770.

C. 1078.

D. 1225.

2. Calculate the GCD of 102947526 and 239821932 using Euclidean algorithm..

A. 11.

B. 12.

C. 8.

D. 6.

3. Calculate the GCD of 8376238 and 1921023 using Euclidean algorithm..

A. 13.

B. 12.

C. 17.

D. 7.

4. What is 11 mod 7 and -11 mod 7?.

A. 4 and 5.

B. 4 and 4.

C. 5 and 3.

D. 4 and -4.

5. Which of the following is a valid property for concurrency?.

A. a = b (mod n) if n.

B. (a-b).

C. a = b (mod n) implies b = a (mod n).

D. a = b (mod n) and b = c (mod n) implies a = c (mod n).

6. [(a mod n) + (b mod n)] mod n = (a+b) mod n.

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

7. [(a mod n) – (b mod n)] mod n = (b – a) mod n.

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

8. 117 mod 13 =

A. 3.

B. 7.

C. 5.

D. 15.

9. The multiplicative Inverse of 1234 mod 4321 is.

A. 3239.

B. 3213.

C. 3242.

D. Does not exist.

10. The multiplicative Inverse of 550 mod 1769 is.

A. 434.

B. 224.

C. 550.

D. Does not exist.

11. The multiplicative Inverse of 24140 mod 40902 is.

A. 2355.

B. 5343.

C. 3534.

D. Does not exist.

12. (6x2 + x + 3)x(5x2 + 2) in Z_10 =

A. x3 + 2x + 6.

B. 5x3 + 7x2 + 2x + 6.

C. x3 + 7x2 + 2x + 6.

D. None of the mentioned.

Answer= 5x3 + 7x2 + 2x + 6

13. Is x3 + 1 reducible over GF(2).

A. Yes.

B. No.

C. Can't Say.

D. Insufficient Data.

14. Is x3 + x2 + 1 reducible over GF(2).

A. Yes.

B. No.

C. Can't Say.

D. Insufficient Data.

15. Is x4 + 1 reducible over GF(2).

A. Yes.

B. No.

C. Can't Say.

D. Insufficient Data.

16. The result of (x2 ? P), and the result of (x ? (x ? P)) are the same, where P is a polynomial..

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

17. The GCD of x3+ x + 1 and x2 + x + 1 over GF(2) is.

A. 1.

B. x + 1.

C. x2.

D. x2 + 1.

18. The GCD of x5+x4+x3 – x2 – x + 1 and x3 + x2 + x + 1 over GF(3) is.

A. 1.

B. x.

C. x + 1.

D. x2 + 1.

19. The GCD of x3 – x + 1 and x2 + 1 over GF(3) is.

A. 1.

B. x.

C. x + 1.

D. x2 + 1.

20. Find the 8-bit word related to the polynomial x6 + x + 1.

A. 1000011.

B. 1000110.

C. 10100110.

D. 11001010.

21. If f(x)=x7+x5+x4+x3+x+1 and g(x)=x3+x+1, find f(x) + g(x)..

A. x7+x5+x4.

B. x7+x5+x4+x3+x.

C. x4+x2+x+1.

D. x6+x4+x2+x+1.

22. If f(x)=x7+x5+x4+x3+x+1 and g(x)=x3+x+1, find f(x) x g(x)..

A. x12+x5+x3+x2+x+1.

B. x10+x4+1.

C. x10+x4+x+1.

D. x7+x5+x+1.

23. If f(x)=x7+x5+x4+x3+x+1 and g(x)=x3+x+1, find the quotient of f(x) / g(x)..

A. x4+x3+1.

B. x4+1.

C. x5+x3+x+1.

D. x3+x2.

24. Primitive Polynomial is also called a ____i) Perfect Polynomialii) Prime Polynomialiii) Irreducible Polynomialiv) Imperfect Polynomial.

A. ii) and iii).

B. only iii).

C. iv) and ii).

D. None.

25. Which of the following are irreducible polynomials?i) X4+X3ii) 1iii) X2+1iv) X4+X+1.

A. i) and ii).

B. only iv).

C. ii) iii) and iv).

D. All of the options.

26. The polynomial f(x)=x3+x+1 is a reducible..

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

27. Find the HCF/GCD of x6+x5+x4+x3+x2+x+1 and x4+x2+x+1..

A. x4+x3+x2+1.

B. x3+x2+1.

C. x2+1.

D. x3+x2+1.

28. On multiplying (x5 + x2 + x) by (x7 + x4 + x3 + x2 + x) in GF(28) with irreducible polynomial (x8 + x4 + x3 + x + 1) we get.

A. x12+x7+x2.

B. x5+x3+x3.

C. x5+x3+x2+x.

D. x5+x3+x2+x+1.

29. On multiplying (x6+x4+x2+x+1) by (x7+x+1) in GF(28) with irreducible polynomial (x8 + x4 + x3 + x + 1) we get.

A. x7+x6+ x3+x2+1.

B. x6+x5+ x2+x+1.

C. x7+x6+1.

D. x7+x6+x+1.

30. Find the inverse of (x2 + 1) modulo (x4 + x + 1)..

A. x4+ x3+x+1.

B. x3+x+1.

C. x3+ x2+x.

D. x2+x.

31. Find the inverse of (x5) modulo (x8+x4 +x3+ x + 1)..

A. x5+ x4+ x3+x+1.

B. x5+ x4+ x3.

C. x5+ x4+ x3+1.

D. x4+ x3+x+1.

32. A very common field in this category is GF(2) with the set {1, 2} and two operations, addition and multiplication..

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

33. Multiplication / Division follow which operation?.

A. XOR.

B. NAND.

C. AND.

D. OR.

34. What do the above numbers correspond to?0 1 2 3 40 4 3 2 10 1 2 3 4– 1 3 2 4.

B. Both Multiplicative Inverses.

C. Additive and Multiplicative Inverse respectively.

D. Multiplicative and Additive Inverses respectively.

35. How many numbers cannot be used in GF(p) in 2n where n=4?.

A. 2.

B. 5.

C. 3.

D. 1.

36. If f(x)=x3+x2+2 and g(x)=x2-x+1, find: f(x) + g(x).

A. x3+2x2-x+3.

B. x3+x2+3.

C. x3+x+1.

D. x2+2x+4.

37. If f(x)=x3+x2+2 and g(x)=x2-x+1, find: f(x) – g(x).

A. x3+x+4.

B. x3+x+1.

C. x3+x2+3.

D. x3+3x+2.

38. If f(x)=x4+x3+2 and g(x)=x3-x+6, find: f(x) + g(x).

A. 2x4+2x3+x+8.

B. x4+2x3-x+8.

C. x4+x2+x+8.

D. x4+x3+8.

39. If f(x)=x4+x2-x+2 and g(x)=x2-x+1, find: f(x) – g(x).

A. x4+1.

B. x2+1.

C. x2+2x+6.

D. x4-1.

40. If f(x)=x3+x2+2 and g(x)=x2-x+1, find the quotient of  f(x) / g(x).

A. x+3.

B. x2+4.

C. x.

D. x+2.

41. If f(x)=x3+x2+2 and g(x)=x2-x+1, find: f(x) x g(x).

A. x4+x2+2x+2.

B. x5+2x3+2x+3.

C. x5+3x2-2x+2.

D. x4+x2+x+1.

42. Find the 8-bit word related to the polynomial x5 + x2 + x.

A. 10011.

B. 1000110.

C. 100110.

D. 11001010.

43. Find the 8-bit word related to the polynomial x6 + x5 + x2 + x +1.

A. 10011.

B. 11000110.

C. 100110.

D. 1100111.

44. If f(x)=x7+x5+x4+x3+x+1 and g(x)=x3+x+1, find f(x) – g(x)..

A. x7+x5+x4+x3.

B. x6+x4+x2+x.

C. x4+x2+x+1.

D. x7+x5+x4.

45. 5/3 mod 7 =

A. 2.

B. 3.

C. 4.

D. 5.

46. The polynomial x4+1 can be represented as

A. (x+1)(x3+x2+1).

B. (x+1)(x3+x2+x).

C. (x)(x2+x+1).

D. None of the mentioned.

47. -5 mod -3 =

A. 3.

B. 2.

C. 1.

D. 5.

48. Multiply the polynomials P1 = x5 +x2+ x) by P2 = (x7 + x4 +x3+x2 + x) in GF(28) with irreducible polynomial (x8 + x4 + x3 + x + 1). The result is.

A. x4+ x3+ x+1.

B. x5+ x3+x2+x+1.

C. x5+ x4+ x3+x+1.

D. x5+ x3+x2+x.

49. Multiply 00100110 by 10011110 in GF(2^8) with modulus 100011011.The result is.

A. 101111.

B. 101100.

C. 1110011.

D. 11101111.

50. Find the inverse of (x7+x+1) modulo (x8 + x4 + x3+ x + 1)..

A. x7+x.

B. x6+x3.

C. x7.

D. x5+1.

51. 7x = 6 mod 5. Then the value of x is.

A. 2.

B. 3.

C. 4.

D. 5.

52. The product of monic polynomials is monic..

A. TRUE.

B. FALSE.

C. Can't Say.

D. None of the mentioned.

53. The product of polynomials of degrees m and n has a degree m+n+1..

A. TRUE.

B. FALSE.

C. Can't Say.

D. None of the mentioned.

54. The sum of polynomials of degrees m and n has degree max[m,n]..

A. TRUE.

B. FALSE.

C. Can't Say.

D. None of the mentioned.

55. (7x + 2)-(x2 + 5) in Z_10 =

A. 9x2 + 7x + 7.

B. 9x2+ 6x + 10.

C. 8x2 + 7x + 6.

D. None of the mentioned.

Answer= 9x2 + 7x + 7

56. GCD(a,b) = GCD(b,a mod b).

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

57. All groups satisfy properties.

A. G-i to G-v.

B. G-i to G-iv.

C. G-i to R-v.

D. R-i to R-v.

58. An Abelian Group satisfies the properties.

A. G-i to G-v.

B. G-i to R-iv.

C. G-i to R-v.

D. R-i to R-v.

59. A Ring satisfies the properties.

A. R-i to R-v.

B. G-i to G-iv.

C. G-i to R-v.

D. G-i to R-iii.

60. A Ring is said to be commutative if it also satisfies the property.

A. R-vi.

B. R-v.

C. R-vii.

D. R-iv.

61. An 'Integral Domain' satisfies the properties.

A. G-i to G-iii.

B. G-i to R-v.

C. G-i to R-vi.

D. G-i to R-iii.

62. A Field satisfies all the properties above from G-i to R-vi..

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

63. In modular arithmetic : (a/b) = b(a^-1).

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

64. a.(b.c) = (a.b).c is the representation for which property?.

A. G-ii.

B. G-iii.

C. R-ii.

D. R-iii.

65. a(b+c) = ac+bc is the representation for which property?.

A. G-ii.

B. G-iii.

C. R-ii.

D. R-iii.

66. For the group Sn of all permutations of n distinct symbols, what is the number of elements in Sn?.

A. n.

B. n-1.

C. 2n.

D. n!.

67. For the group Sn of all permutations of n distinct symbols, Sn is an abelian group for all values of n..

A. TRUE.

B. FALSE.

C. Nothing can be said.

D. None of the mentioned.

68. Does the set of residue classes (mod 3) form a group with respect to modular addition?.

A. Yes.

B. No.

C. Can't Say.

D. Insufficient Data.