Modular Arithmetic: Master Number Theory and Cryptography Concepts

Question 1

Which property is NOT fundamental to modular arithmetic?

Accepted Answer

Existence of modular multiplicative inverses for all non-zero elements

Answer

Closure under modular addition and multiplication

Answer

Commutativity and associativity of modular addition and multiplication

Answer

Distributivity of modular multiplication over modular addition

Question 2

Which of the following is a practical application of modular arithmetic?

Accepted Answer

Cryptography

Answer

Calculus

Answer

Algebra

Answer

Geometry

Question 3

Which of the following statements about congruence modulo a positive integer n is true?

Accepted Answer

If a is congruent to b modulo n and b is congruent to c modulo n, then a is congruent to c modulo n.

Answer

If a is congruent to b modulo n and b is congruent to c modulo n, then a is congruent to b and b is congruent to c.

Answer

If a is congruent to b modulo n and b is congruent to c modulo n, then a is congruent to c but not necessarily b is congruent to c.

Question 4

Find the smallest positive integer n such that 5n is congruent to 1 modulo 12.

Accepted Answer

11

Answer

13

Answer

10

Answer

12

Question 5

Find the remainder when 23,456 is divided by 7.

Accepted Answer

1

Answer

2

Answer

4

Answer

3

Question 6

Which of the following is NOT a property of modular arithmetic?

Accepted Answer

Closure under subtraction

Answer

Closure under addition

Answer

Associativity under addition

Answer

Commutativity under addition

Question 7

Which of the following is the most common application of modular arithmetic?

Accepted Answer

Cryptography

Answer

Timekeeping

Answer

Financial accounting

Answer

Calendar calculations

Question 8

An RSA encryption system has a public key (n, e) = (443, 7). What is the private key?

Accepted Answer

(443, 157)

Answer

(443, 159)

Answer

(443, 158)

Answer

(443, 156)

Question 9

Which of the following is **not an application** of modular arithmetic?

Accepted Answer

Prime number factorization

Answer

Error correction

Answer

Public-key cryptography

Answer

Checksums

Question 10

If a ≡ b (mod n) and b ≡ c (mod n), then which of the following is true?

Accepted Answer

a ≡ c (mod n)

Answer

a > c (mod n)

Answer

a < c (mod n)

Answer

a / c (mod n)

Question 11

Which of the following is **not** a property of modular arithmetic?

Accepted Answer

Closure under subtraction

Answer

Closure under addition

Answer

Associativity of addition

Answer

Commutativity of addition

Question 12

Solve the congruence equation: 3x ≡ 2 (mod 7).

Accepted Answer

x ≡ 5 (mod 7)

Answer

x ≡ 4 (mod 7)

Answer

x ≡ 6 (mod 7)

Answer

x ≡ 2 (mod 7)

Question 13

Find the remainder when 2^1000 is divided by 5.

Accepted Answer

1

Answer

3

Answer

2

Answer

4

Question 14

What is the remainder when 2023 is divided by 7?

Accepted Answer

5

Answer

6

Answer

3

Answer

2

Question 15

Which of the following is congruent to 3 modulo 5?

Accepted Answer

x ≡ 8 (mod 5)

Answer

x ≡ 4 (mod 5)

Answer

x ≡ 6 (mod 5)

Answer

x ≡ 9 (mod 5)

Question 16

If a ≡ 2 (mod 5) and b ≡ 3 (mod 5), what is (a + b) congruent to modulo 5?

Accepted Answer

(a + b) ≡ 0 (mod 5)

Answer

(a + b) ≡ 2 (mod 5)

Answer

(a + b) ≡ 1 (mod 5)

Question 17

Calculate the multiplicative inverse of 3 modulo 7.

Accepted Answer

5

Answer

2

Answer

6

Answer

4

Question 18

What is the smallest positive integer solution for x in the congruence equation: 2x ≡ 4 (mod 6)?

Accepted Answer

x ≡ 2 (mod 6)

Answer

x ≡ 4 (mod 6)

Answer

x ≡ 5 (mod 6)

Answer

x ≡ 1 (mod 6)

Question 19

Which of the following is a possible unit digit of 7^n, where n is a positive integer?

Accepted Answer

9

Answer

4

Answer

2

Answer

6

Question 20

In the RSA encryption algorithm, the public key (e, n) consists of two numbers. What property should 'e' satisfy with respect to φ(n), where φ is Euler's totient function??

Accepted Answer

e and φ(n) are relatively prime.

Answer

e is a prime number.

Answer

e is even.

Answer

e is a divisor of n.

Question 21

What is the remainder when 3^100 is divided by 4?

Accepted Answer

1

Answer

3

Answer

0

Answer

2

Question 22

If a number is divisible by 6, then it must also be divisible by:

Accepted Answer

Both 2 and 3

Answer

12

Answer

4

Answer

9

Question 23

In a modulo 12 system, what time is it 5 hours after 10 o'clock?

Accepted Answer

3 o'clock

Answer

5 o'clock

Answer

7 o'clock

Question 24

What is the remainder when 12345 is divided by 7?

Accepted Answer

3

Answer

2

Answer

5

Answer

1

Question 25

Which of the following is congruent to 17 modulo 5?

Accepted Answer

2

Answer

12

Answer

7

Answer

3

Question 26

If a ≡ b (mod m), which of the following is always true?

Accepted Answer

a - b is divisible by m

Answer

a / b is divisible by m

Answer

a + b is divisible by m

Answer

a * b is divisible by m

Question 27

Find the multiplicative inverse of 3 modulo 7.

Accepted Answer

5

Answer

2

Answer

4

Answer

6

Question 28

Solve the linear congruence 3x ≡ 2 (mod 5).

Accepted Answer

x ≡ 4 (mod 5)

Answer

x ≡ 3 (mod 5)

Answer

x ≡ 2 (mod 5)

Answer

x ≡ 1 (mod 5)

Question 29

What is the value of 2^10 (mod 11)?

Accepted Answer

1

Answer

10

Answer

2

Answer

3

Question 30

Which of the following is a Fermat prime?

Accepted Answer

17

Answer

25

Answer

21

Answer

15

Question 31

A message is encrypted using the RSA algorithm with public key (n, e) = (15, 7). What is the corresponding private key (d)?

Accepted Answer

13

Answer

7

Answer

11

Answer

5

Question 32

Which of the following is a necessary condition for a number to be divisible by 9?

Accepted Answer

The sum of its digits is divisible by 9.

Answer

The number is odd.

Answer

The number is even.

Answer

The number ends in a 0.

Question 33

In a group of 100 people, 60 like apples, 50 like oranges, and 20 like both. How many people like neither apples nor oranges?

Accepted Answer

30

Answer

20

Answer

10

Answer

40

Question 34

What is the remainder when 12345 is divided by 7?

Accepted Answer

3

Answer

1

Answer

4

Answer

2

Question 35

Which of the following is congruent to 17 modulo 5?

Accepted Answer

2

Answer

3

Answer

7

Answer

4

Question 36

If a ≡ b (mod m) and c ≡ d (mod m), which of the following is always true?

Accepted Answer

a + c ≡ b + d (mod m)

Answer

a * c ≡ b * d (mod m)

Answer

a / c ≡ b / d (mod m)

Answer

a - c ≡ b - d (mod m)

Question 37

What is the multiplicative inverse of 3 modulo 7?

Accepted Answer

5

Answer

4

Answer

6

Answer

2

Question 38

Solve the linear congruence 2x ≡ 5 (mod 7).

Accepted Answer

x ≡ 6 (mod 7)

Answer

x ≡ 1 (mod 7)

Answer

x ≡ 3 (mod 7)

Answer

x ≡ 5 (mod 7)

Question 39

Which of the following is a solution to the system of congruences: x ≡ 1 (mod 3) and x ≡ 2 (mod 5)?

Accepted Answer

x ≡ 11 (mod 15)

Answer

x ≡ 2 (mod 15)

Answer

x ≡ 8 (mod 15)

Answer

x ≡ 7 (mod 15)

Question 40

What is the order of the element 3 modulo 7?

Accepted Answer

6

Answer

2

Answer

3

Answer

4

Question 41

In a group of 12 people, how many ways can they be arranged in a circle if rotations are considered the same arrangement?

Accepted Answer

11!

Answer

12! / 2

Answer

12! / 12

Answer

12!

Question 42

The RSA cryptosystem uses which of the following mathematical concepts?

Accepted Answer

Modular arithmetic

Answer

Linear algebra

Answer

Calculus

Answer

Differential equations

Question 43

If a = 15 and b = 2, what is the value of a mod b?

Accepted Answer

1

Answer

0

Answer

3

Answer

2