gs gs124 More Number Theory - Number Theory - Quiz No.1

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Question 1: What is the Discrete logarithm to the base 10 (mod 19) for a =7?

12

14

8

11

Question 2: 3²⁰¹ mod 11 =

3

5

6

10

Question 3: Find a number x between 0 and 28 with x^85 congruent to 6 mod 29.

22

12

6

18

Question 4: What is the Discrete logarithm to the base 13 (mod 19) for a =13?

14

1

8

17

Question 5: What is the Discrete logarithm to the base 15 (mod 19) for a =9?

3

7

12

4

Question 6: Find a number x between 0 and 28 with x⁸⁵ congruent to 6 mod 35.

6

32

8

28

Question 7: Find a number ‘a’ between 0 and 72 with ‘a’ congruent to 9794 mod 73.

53

29

12

37

Question 8: What is the Discrete logarithm to the base 2 (mod 19) for a =7?

3

4

6

9

Question 9: ᶲ(41)=

40

20

18

22

Question 10: ᶲ(27)=

6

12

26

18

Question 11: Find a number ‘a’ between 0 and 9 such that ‘a’ is congruent to 7^1000 mod 10.

2

1

3

4

Question 12: ᶲ(440)=

200

180

160

220

Question 13: GCD(n,n+1) = 1 always.

True

False

Question 14: Find the order of the group G = <Z12*, ×>?

4

5

6

2

Question 15: Find the order of the group G = <Z21*, ×>?

12

8

13

11

Question 16: Find the order of group G= <Z20*, x>

6

9

10

8

Question 17: Find the order of group G= <Z7*, x>

6

4

3

5

Question 18: 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.

True

False

Question 19: In the order of group G= <Z20*, x>, what is the order of element 17?

16

4

11

6

Question 20: The order of group G= <Z9, x> , primitive roots of the group are –

8 , Primitive roots- 2,3

6 , Primitive roots- 5

6 , Primitive roots- 2,5

6 , Primitive roots- 5,7

Question 21: Which among the following values: 17, 20, 38, and 50, does not have primitive roots in the group G = <Zn*, ×>?

17

20

38

50

Question 22: Find the number of primitive roots of G=<Z11*, x>?

5

6

4

10

Question 23: Find the primitive roots of G=<Z11*, x>?.

{2, 6, 8}

{2, 5, 8}

{3, 4, 7, 8}

{2, 6, 7, 8}

Question 24: If a group has primitive roots, it is a cyclic group

True

False

Question 25: Find the primitive roots of G = <Z10*, ×>.

{2, 6, 8}

{3,6 ,9}

{3, 7, 8}

{3, 7}

Question 26: The group G = <Zp*, ×> is always cyclic.

True

False

Question 27: gcd( 18,300) =

4

12

8

6

Question 28: ᶲ(37)=

24

22

13

36

Question 29: ᶲ(35)=

24

25

22

18

Question 30: ᶲ(21)=

10

12

8

14

Question 31: 7³ mod 19 =

18

1

14

12

Question 32: 7^(3+j) mod 19 =

7^j mod 19

1 mod 19

7³ + 7^j mod 19

All of the mentioned are true

Question 33: What is the period of 7^m mod 19?

2

3

4

5

Question 34: ᶲ(19)=

14

13

18

17

Question 35: What is the period of 11 (mod 19)

2

3

4

5

Question 36: What is the period of 17 (mod 19)

5

7

9

11

Question 37: What is the period of 9 (mod 19)

12

10

11

9

Question 38: How many primitive roots does Z<19> have?

5

8

7

6

Question 39: Which of the below properties are correct?
Consider the following Logarithmic Properties –
i) y = x^(log_x(y))
ii) log_x(1) = 1
iii) log_x(x) = 0
iv) log_x(yz) = log_x(y) + log_x(z)
v) log_x(y^r) – r x log_x(y)

1st 2nd and 4th

2nd 3rd and 5th

2nd 4th and 5th

1st 4th and 5th

Question 40: What is the Discrete logarithm to the base 10 (mod 19) for a =7?

12

14

8

11

Question 41: 3²⁰¹ mod 11 =

3

5

6

10

Question 42: Find a number x between 0 and 28 with x^85 congruent to 6 mod 29.

22

12

6

18

Question 43: What is the Discrete logarithm to the base 13 (mod 19) for a =13?

14

1

8

17

Question 44: What is the Discrete logarithm to the base 15 (mod 19) for a =9?

3

7

12

4

Question 45: Find a number x between 0 and 28 with x⁸⁵ congruent to 6 mod 35.

6

32

8

28

Question 46: Find a number ‘a’ between 0 and 72 with ‘a’ congruent to 9794 mod 73.

53

29

12

37

Question 47: What is the Discrete logarithm to the base 2 (mod 19) for a =7?

3

4

6

9

Question 48: ᶲ(41)=

40

20

18

22

Question 49: ᶲ(27)=

6

12

26

18

Question 50: Find a number ‘a’ between 0 and 9 such that ‘a’ is congruent to 7^1000 mod 10.

2

1

3

4

Question 51: ᶲ(440)=

200

180

160

220

Question 52: GCD(n,n+1) = 1 always.

True

False

Question 53: ᶲ(231)=

230

60

80

120

Question 54: n is prime if and only if n divides (2ⁿ – 2).

True

False

Question 55: Find x for the CRT when x= 2 mod 3; x= 3 mod 5; x = 2 mod 7

33

22

23

31

Question 56: Consider a function: f(n) = number of elements in the set {a: 0 <= a < n and gcd(a,n) = 1}. What is this function?

Primitive

Totient

Primality

All of the mentioned

Question 57: The inverse of 49 mod 37 is –

31

23

22

34

Question 58: 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.

354

371

432

213

Question 59: How many primitive roots are there for 25?

4

5

7

8

Question 60: 17 x² = 10 ( mod 29 )

x = 3, 22 (mod 29)

x = 7, 28 (mod 29)

x = 2, 27 (mod 29)

x = 4, 28 (mod 29)

Question 61: x – 4x – 16 = 0 (mod 29)

x = 6, 24 (mod 29)

x = 9, 24 (mod 29)

x = 9, 22 (mod 29)

x = 6, 22 (mod 29)

Question 62: x⁷ = 17 (mod 29)

x = 8, 9, 12, 13, 15, 24, 28 (mod 29)

x = 8, 10, 12, 15, 18, 26, 27 (mod 29)

x = 8, 10, 12, 15, 17, 24, 27 (mod 29)

x = 8, 9, 13, 15, 17, 24, 28 (mod 29)

Question 63: The inverse of 37 mod 49 is –

23

12

4

6

Question 64: How many primitive roots are there for 19?

4

5

3

6

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