Number Theory Quizzes: Test Your Skills in Number Theory

Question 1

Regarding primitive roots, which statement is true?

Accepted Answer

For a prime number p, g is its primitive root if and only if the greatest common divisor (gcd) of g and p is 1.

Answer

Every positive integer has a primitive root.

Answer

A positive integer has a primitive root only if it's prime.

Answer

The number of primitive roots for a positive integer is always even.

Question 2

In modular arithmetic, what does the statement 'a ≡ b (mod c)' indicate about the relationship between integers a, b, and c?

Accepted Answer

a and b have the same remainder when divided by c.

Answer

a and b are both even numbers.

Answer

a and b are both odd numbers.

Answer

a and b are both divisible by c.

Question 3

An integer n is divisible by both 4 and 6. Which of the following must be true?

Accepted Answer

n is divisible by 12.

Answer

n is divisible by 8.

Answer

n is divisible by 9.

Question 4

Which of the following is a defining characteristic of a prime number?

Accepted Answer

It has only two factors, 1 and itself.

Answer

It is divisible by 2.

Answer

The sum of its digits is an odd number.

Question 5

In modular arithmetic, consider the equation x^2 ≡ a (mod p), where p is prime. Which statement is correct?

Accepted Answer

There's exactly one solution if a is a quadratic residue modulo p.

Answer

There's a unique solution for any a.

Answer

There are always two solutions if a is not a quadratic residue modulo p.

Answer

There are no solutions if p is an odd prime.

Question 6

Let p be an odd prime number and a be an integer. Determine the value of $\left(\frac{a}{p}\right)$ if $$\left(\frac{2a}{p}\right)=1$$

Accepted Answer

$\left(\frac{a}{p}\right)=1$

Answer

$\left(\frac{a}{p}\right)=-1$

Answer

$\left(\frac{a}{p}\right)=0$

Question 7

What is a necessary condition for two integers a and b to be relatively prime?

Accepted Answer

a and b have no common factors other than 1.

Answer

a and b are both odd.

Answer

a and b are both perfect squares.

Question 8

In Fermat's Little Theorem, for any prime number (p) and any integer (a), what is **always** true?

Accepted Answer

(a^p = 1 mod p)

Answer

(a^p = a^2 mod p)

Answer

(a^p = a^-1 mod p)

Answer

(a^p = 0 mod p)

Question 9

Which of the following is NOT a necessary condition for 'a' to be a primitive root modulo a prime number 'p'?

Accepted Answer

a^(p-1) is divisible by p

Answer

a is coprime to p

Answer

a^2 is not divisible by p

Question 10

In the Baby-Step Giant-Step algorithm, what is the time complexity of the precomputation step, where we create the table of powers?

Accepted Answer

O(√p)

Answer

O(p)

Answer

O(1)

Answer

O(p^2)

Question 11

Identify the condition that is both necessary and sufficient for a positive integer n to be a perfect square:

Accepted Answer

The prime factorization of n has even exponents for all prime factors.

Answer

n is divisible by a prime number.

Answer

n is the sum of two consecutive prime numbers.

Question 12

Prove that a positive integer n can be uniquely expressed as a product of primes if and only if n has a unique prime factorization.

Accepted Answer

True.

Answer

False.

Answer

True, but the converse is false.

Answer

False, but the converse is true.

Question 13

Which of the following is a composite number?

Accepted Answer

9

Answer

2

Answer

3

Answer

5

Question 14

Which of the following is a perfect cube?

Accepted Answer

27

Answer

8

Answer

16

Answer

36

Question 15

What is the order of the group (Z/12Z)*?

Accepted Answer

6

Answer

4

Answer

10

Answer

12

Question 16

Classify the number 97 based on its properties.

Accepted Answer

Composite number

Answer

Relatively prime number

Answer

Perfect square

Answer

Perfect cube

Question 17

State the necessary and sufficient condition for a number to be divisible by 3.

Accepted Answer

The sum of its digits is divisible by 3.

Answer

The last digit is divisible by 3.

Answer

The sum of the first and last digits is divisible by 3.

Answer

The number is prime.

Question 18

Which integer among the following options is a prime number?

Accepted Answer

17

Answer

12

Answer

25

Answer

33

Question 19

Determine if 105 is divisible by 7:

Accepted Answer

Yes

Answer

No

Answer

Cannot be determined

Question 20

In modern cryptography, what is a key advantage of using elliptic curves?

Accepted Answer

Elliptic curves enhance the security of key exchange protocols because solving the elliptic curve discrete logarithm problem is difficult.

Answer

Elliptic curves are primarily used to generate random numbers for cryptography.

Answer

Elliptic curves offer significant speed advantages in encryption and decryption compared to other cryptographic methods.

Question 21

Prove that the sum of any two consecutive odd numbers is even.

Accepted Answer

Use the definition of consecutive odd numbers and the properties of even and odd numbers

Answer

Use mathematical induction

Answer

Use the Euclidean algorithm

Answer

Use modular arithmetic

Question 22

Which of the following is not a perfect number?

Accepted Answer

33

Answer

6

Answer

28

Answer

8128

Question 23

Find the greatest common divisor (GCD) of 24 and 36.

Accepted Answer

6

Answer

1

Answer

2

Answer

12

Question 24

If p is a prime number and a is a primitive root modulo p, which of the following properties is true?

Accepted Answer

a^(p-1) ≡ 1 (mod p)

Answer

a^p-1 ≡ 1 (mod p)

Answer

a^p ≡ 1 (mod p)

Answer

a^(p-2) ≡ 1 (mod p)

Question 25

Find the number of solutions to the congruence x^2 ≡ 1 (mod 8).

Accepted Answer

2

Answer

1

Answer

4

Answer

8

Question 26

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

Accepted Answer

It has no divisors other than itself and 1

Answer

It is greater than 1

Answer

It has exactly two distinct factors

Answer

It is odd

Question 27

Find the greatest common divisor (GCD) of 24 and 36:

Accepted Answer

6

Answer

2

Answer

3

Answer

12

Question 28

Find the greatest common divisor (GCD) of 12 and 18.

Accepted Answer

3

Answer

2

Answer

6

Answer

12

Question 29

Which of the following is the fundamental theorem of arithmetic?

Accepted Answer

Every integer can be expressed as a product of prime numbers (unique factors that cannot be further broken down).

Answer

Every prime number is odd.

Answer

The sum of two primes is always prime.

Answer

The GCD of two coprime numbers (numbers that share no common factors other than 1) is 1.

Question 30

If p is a prime number and p divides ab (p is a factor of ab), then p divides a or p divides b.

Accepted Answer

True

Answer

False

Question 31

Which of the following algorithms is used to find the GCD of two integers?

Accepted Answer

Euclidean algorithm (a method for computing the greatest common divisor of two integers)

Answer

Fermat's little theorem (a theorem in number theory that states that if p is a prime number, then for any integer a, a^p ≡ a (mod p))

Answer

Chinese remainder theorem (a theorem in number theory that provides a way to solve a system of simultaneous linear congruences)

Answer

Sieve of Eratosthenes (a method for finding all prime numbers up to a certain limit)

Question 32

Which of the following is a prime number?

Accepted Answer

19

Answer

12

Answer

25

Answer

36

Question 33

Find the smallest positive integer x such that x ≡ 5 (mod 11) and x ≡ 3 (mod 13).

Accepted Answer

23

Answer

65

Answer

87

Answer

109

Question 34

If a and b are relatively prime, then their greatest common factor (GCF) is:

Accepted Answer

1

Answer

a + b

Answer

a - b

Answer

a * b

Question 35

If p is a prime number, the equation x^2 ≡ 1 (mod p) has:

Accepted Answer

Exactly two solutions

Answer

No solutions

Answer

Exactly one solution

Answer

More than two solutions

Question 36

Which of the following is a prime number?

Accepted Answer

17

Answer

12

Answer

24

Answer

36

Question 37

Find the number of positive divisors of 24.

Accepted Answer

6

Answer

3

Answer

4

Answer

8

Question 38

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

Accepted Answer

They are even.

Answer

They have exactly two distinct positive factors.

Answer

They are greater than 1.

Answer

They are divisible only by themselves and 1.

Question 39

Find the number of positive integers less than 100 that are divisible by both 4 and 6.

Accepted Answer

8

Answer

20

Answer

15

Answer

5

Question 40

Which of the following statements about Fermat's Little Theorem is true?

Accepted Answer

If p is a prime number and a is an integer not divisible by p, then a^(p-1) ≡ 1 (mod p).

Answer

If p is a prime number and a is any integer, then a^p ≡ a (mod p).

Answer

If p is a composite number and a is an integer not divisible by p, then a^(p-1) ≡ 0 (mod p).

Question 41

Which of the following is an application of number theory in computer science?

Accepted Answer

Public-key cryptography

Answer

Calculus

Answer

Statistics

Answer

Linear programming

Question 42

Identify the prime number among the given options:

Accepted Answer

17

Answer

32

Answer

46

Answer

10

Question 43

Determine the smallest positive integer that is divisible by both 12 and 18:

Accepted Answer

36

Answer

60

Answer

24

Answer

48

Question 44

Which of the following is the greatest common divisor (GCD) of 18 and 24?

Accepted Answer

6

Answer

12

Answer

9

Question 45

Find the multiplicative inverse of 5 modulo 13.

Accepted Answer

12

Answer

5

Answer

8

Answer

4

Question 46

Which of the following is a necessary and sufficient condition for an integer 'n' to be divisible by 3?

Accepted Answer

The sum of the digits of 'n' is divisible by 3.

Answer

'n' is odd.

Answer

'n' is a perfect square.

Answer

'n' is a multiple of 6.

Question 47

What is the greatest common divisor (GCD) of 120 and 168?

Accepted Answer

24

Answer

48

Answer

12

Question 48

If 'a' and 'b' are integers such that 'a' is congruent to 'b' modulo '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 49

What is the remainder when 100! is divided by 101?

Accepted Answer

100

Answer

1

Answer

50

Answer

0

Question 50

Which of the following is a prime number?

Accepted Answer

19

Answer

25

Answer

9

Answer

1

Question 51

What is the least common multiple (LCM) of 15 and 24?

Accepted Answer

120

Answer

60

Answer

30

Answer

360

Question 52

The equation 2^x = 3 has a solution in the set of...

Accepted Answer

Real numbers

Answer

Natural numbers

Answer

Integers

Answer

Rational numbers

Question 53

If 'a' and 'b' are relatively prime integers, what is the GCD of 'a' and 'b'?

Accepted Answer

1

Answer

a * b

Answer

b

Answer

a

Question 54

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

Accepted Answer

40

Answer

20

Answer

60

Answer

80

Question 55

The sum of the squares of the first 'n' positive integers is given by...

Accepted Answer

n(n+1)(2n+1)/6

Answer

n(n+1)/2

Answer

n^2(n+1)^2/4

Answer

n(n+1)(n+2)/3