Prime Numbers: Definition, Properties, and Applications

Question 1

What is the smallest prime number?

Accepted Answer

2

Answer

5

Answer

3

Answer

1

Question 2

State the Goldbach conjecture.

Accepted Answer

Every even integer greater than 2 can be expressed as the sum of two prime numbers.

Answer

Every odd integer greater than 3 can be expressed as the sum of three prime numbers.

Answer

Every prime number greater than 3 can be expressed as the sum of two prime numbers.

Question 3

Which of the following applications heavily relies on the concept of prime numbers?

Accepted Answer

Cryptography

Answer

Database management

Answer

Statistics

Answer

Computer graphics

Question 4

Which of the following sequences consists exclusively of prime numbers?

Accepted Answer

2, 3, 5, 7, 11, ...

Answer

1, 3, 5, 7, 9, ...

Answer

1, 2, 3, 4, 5, ...

Answer

2, 4, 6, 8, 10, ...

Question 5

Explain the significance of prime numbers in public-key cryptography.

Accepted Answer

They are used to generate large prime numbers that form the foundation of secure communication protocols.

Answer

They are used to encrypt messages directly.

Answer

They are used to crack secret codes.

Question 6

In cryptography, what is the primary use of prime numbers?

Accepted Answer

Generating public keys for encryption algorithms

Answer

Decrypting private messages without a key

Answer

Creating one-time passwords

Answer

Encoding data for digital signatures

Question 7

Which of the following is a prime number?

Accepted Answer

11

Answer

15

Answer

9

Answer

20

Question 8

Which of the following theorems states that every even number greater than 2 is composite?

Accepted Answer

Euclid's Theorem

Answer

Euler's Theorem

Answer

Wilson's Theorem

Answer

Fermat's Last Theorem

Question 9

What is the fundamental theorem of arithmetic?

Accepted Answer

Every integer greater than 1 can be uniquely expressed as a product of prime numbers.

Answer

Every prime number is odd.

Answer

The product of two prime numbers is always prime.

Question 10

Which of the following is an application of prime numbers in cryptography?

Accepted Answer

RSA encryption

Answer

Calculus

Answer

Linear programming

Answer

Differential equations

Question 11

Which of the following is a necessary and sufficient condition for a number to be prime?

Accepted Answer

It has exactly two distinct positive divisors.

Answer

It is divisible by itself and 1.

Answer

It is odd.

Answer

It is greater than 1.

Question 12

What is the sum of all the prime numbers less than 100?

Accepted Answer

226

Answer

1000

Answer

100

Answer

500

Question 13

Which of the following is a Mersenne prime?

Accepted Answer

2^127 - 1

Answer

2^17 - 1

Answer

2^120 - 1

Answer

2^23 - 1

Question 14

What is the significance of the largest known prime number?

Accepted Answer

It demonstrates the vastness and complexity of prime numbers and provides a benchmark for computational power.

Answer

It is the largest number ever calculated by humans.

Answer

It has no practical applications.

Question 15

What is the smallest **prime** number?

Accepted Answer

2

Answer

5

Answer

1

Answer

3

Question 16

The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a unique product of:

Accepted Answer

Prime numbers

Answer

Perfect numbers

Answer

Even numbers

Answer

Odd numbers

Question 17

The Goldbach conjecture, which is still unproven, suggests that every even integer greater than 2 can be expressed as the sum of:

Accepted Answer

Two prime numbers

Answer

Two Mersenne primes

Answer

Two odd numbers

Answer

Two perfect numbers

Question 18

The Sieve of Eratosthenes is an algorithm used to find all the prime numbers up to a given integer. What is the **first** step in this algorithm?

Accepted Answer

Create a list of all natural numbers up to the given integer.

Answer

Cross out all even numbers greater than 2.

Answer

Find the smallest remaining number and cross out all its multiples.

Question 19

Which of the following is a **sufficient** condition for a number to be prime?

Accepted Answer

It has exactly two distinct factors.

Answer

It is an odd number.

Answer

It is divisible by 2.

Question 20

Which of the following numbers is a prime number?

Accepted Answer

7

Answer

15

Answer

12

Answer

9

Question 21

What is the smallest prime number?

Accepted Answer

2

Answer

0

Answer

-2

Answer

1

Question 22

What is the Goldbach conjecture?

Accepted Answer

Every even integer greater than 2 can be expressed as the sum of two prime numbers.

Answer

Every prime number can be expressed as the sum of two squares.

Answer

There are infinitely many prime numbers.

Question 23

What is the difference between a prime number and a composite number?

Accepted Answer

A prime number has exactly two distinct factors (1 and itself), while a composite number has more than two distinct factors.

Answer

A prime number is always even, while a composite number is always odd.

Question 24

Which of the following is an application of prime numbers in cryptography?

Accepted Answer

RSA encryption

Answer

Solving quadratic equations

Answer

Calculating the volume of a sphere

Answer

Finding the area of a circle

Question 25

What is the Riemann Hypothesis?

Accepted Answer

A conjecture that all non-trivial zeros of the Riemann zeta function have a real part of 1/2.

Answer

A theorem that states that there are infinitely many prime numbers.

Answer

A formula for finding the prime factorization of any number.

Question 26

Which of the following is NOT a characteristic of prime numbers?

Accepted Answer

They are even.

Answer

They are greater than 1.

Answer

They cannot be expressed as a product of two smaller natural numbers.

Answer

They have exactly two factors, 1 and themselves.

Question 27

What is the smallest prime number?

Accepted Answer

2

Answer

0

Answer

3

Answer

1

Question 28

Which of the following numbers is a composite number?

Accepted Answer

12

Answer

7

Answer

17

Answer

13

Question 29

What is the prime factorization of 60?

Accepted Answer

2^2 x 3 x 5

Answer

3 x 4 x 5

Answer

2 x 3 x 10

Answer

2 x 3 x 5 x 5

Question 30

Which of the following theorems states that any integer greater than 1 can be uniquely factorized into a product of prime numbers?

Accepted Answer

Fundamental Theorem of Arithmetic

Answer

Euclid's Lemma

Answer

Goldbach's Conjecture

Answer

Prime Number Theorem

Question 31

Which of the following is a practical application of prime numbers?

Accepted Answer

Cryptography

Answer

Social network analysis

Answer

Medical diagnosis

Answer

Weather forecasting

Question 32

What is the Riemann Hypothesis?

Accepted Answer

A conjecture that all non-trivial zeros of the Riemann zeta function lie on the critical line.

Answer

A conjecture that states that there are infinitely many prime numbers.

Answer

A theorem that states that every odd number greater than 1 can be expressed as the sum of three prime numbers.

Question 33

Which of the following is a characteristic of prime numbers?

Accepted Answer

They have exactly two distinct factors.

Answer

They are divisible by 4 or 6.

Answer

They are even.

Answer

They are all greater than 10.

Question 34

Which of the following is a practical application of prime numbers?

Accepted Answer

Public key cryptography

Answer

Linear regression

Answer

Quantum entanglement

Answer

Calculus

Question 35

What is the name given to the exponent n when 2ⁿ - 1 is a prime number?

Accepted Answer

Mersenne exponent

Answer

Euler exponent

Answer

Wilson exponent

Answer

Fermat exponent

Question 36

Which of the following is a probabilistic test for primality?

Accepted Answer

Miller-Rabin test

Answer

Euclid's algorithm

Answer

Lagrange's theorem

Answer

Fermat's Little Theorem

Question 37

Which of the following is NOT a characteristic of prime numbers?

Accepted Answer

They are even.

Answer

They are the basic building blocks of composite numbers.

Answer

They have exactly two distinct factors.

Question 38

Given the integer 21, what is its smallest prime factor?

Accepted Answer

3

Answer

5

Answer

11

Answer

7

Question 39

A direct consequence of the Fundamental Theorem of Arithmetic is:

Accepted Answer

Every integer greater than 1 can be uniquely factorized into a product of prime numbers.

Answer

The sum of two prime numbers is always a prime number.

Answer

There are infinitely many prime numbers.

Question 40

How many positive integers less than 100 are relatively prime to 100?

Accepted Answer

40

Answer

75

Answer

25

Answer

50

Question 41

The greatest common divisor of 210 and 462 is:

Accepted Answer

105

Answer

70

Answer

35

Answer

15

Question 42

The least common multiple of 12, 18, and 24 is:

Accepted Answer

72

Answer

36

Answer

54

Answer

108

Question 43

Which statement about the distribution of prime numbers is true?

Accepted Answer

The Prime Number Theorem provides an approximation for the number of prime numbers less than a given number.

Answer

Prime numbers are evenly distributed among integers.

Question 44

How many primes between 1 and 1000 are congruent to 1 modulo 4?

Accepted Answer

168

Answer

120

Answer

210

Answer

252

Question 45

In cryptography, a key application of prime numbers is:

Accepted Answer

Public-key cryptography relies on the difficulty of factoring large prime numbers.

Answer

Prime numbers are used to compress data.

Answer

Prime numbers are used to generate pseudorandom numbers.

Question 46

According to the Goldbach conjecture, every even integer greater than 2 can be expressed as the sum of:

Accepted Answer

Two prime numbers

Answer

Two odd numbers

Answer

Four prime numbers

Answer

Three prime numbers

Question 47

Which of the following numbers is a prime number?

Accepted Answer

17

Answer

12

Answer

24

Answer

21

Question 48

What is the smallest prime number?

Accepted Answer

2

Answer

1

Answer

5

Answer

3

Question 49

Which of the following is a composite number?

Accepted Answer

36

Answer

11

Answer

19

Answer

2

Question 50

How many prime numbers are there between 1 and 10 (excluding 1 and 10)?

Accepted Answer

4

Answer

3

Answer

6

Answer

5

Question 51

What is the prime factorization of 48?

Accepted Answer

2<sup>4</sup> * 3

Answer

4 * 12

Answer

2<sup>2</sup> * 12

Answer

2 * 3 * 8

Question 52

Which of the following statements is true about prime numbers?

Accepted Answer

Every integer greater than 1 can be uniquely expressed as a product of prime numbers (Fundamental Theorem of Arithmetic).

Answer

There are only finitely many prime numbers.

Answer

All prime numbers are odd.

Question 53

What is the greatest common divisor (GCD) of 24 and 36?

Accepted Answer

12

Answer

24

Answer

36

Answer

6

Question 54

The RSA algorithm, widely used for encrypting data, relies on the difficulty of:

Accepted Answer

Factoring large integers into their prime factors.

Answer

Performing modular arithmetic.

Answer

Finding the greatest common divisor of two numbers.

Answer

Generating large prime numbers.

Question 55

Which of the following is NOT a prime number?

Accepted Answer

9

Answer

13

Answer

17

Answer

7

Question 56

What is the next prime number after 29?

Accepted Answer

31

Answer

32

Answer

30

Answer

33