Mathematics · Quantitative Aptitude

Number Theory

438 Questions

Number theory explores the properties and relationships of integers, focusing heavily on prime and composite numbers. Key areas include identifying prime numbers, understanding co primes, and exploring mathematical conjectures like Goldbach's. These questions are a core component of quantitative aptitude sections in banking and government exams.

Prime number identificationCo prime propertiesGoldbach conjectureOdd and even numbersDivisibility rules

Number Theory Questions

Multiple choice

What is the name of the conjecture that states that every even number greater than 2 can be written as the sum of two primes?

  1. Goldbach's Conjecture

  2. Fermat's Last Theorem

  3. Riemann Hypothesis

  4. P versus NP
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Goldbach's Conjecture states that every even number greater than 2 can be written as the sum of two primes. This conjecture has been proven for all even numbers up to 4³10^{18}, but it remains unproven in general.

Multiple choice

What is the name of the mathematical theory that studies the properties of prime numbers?

  1. Number theory

  2. Algebraic geometry

  3. Topology

  4. Analysis
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Number theory is a mathematical theory that studies the properties of prime numbers and other integers.

Multiple choice

Which of the following is a famous unsolved problem in Complexity Theory?

  1. P versus NP Problem

  2. Goldbach's Conjecture

  3. Riemann Hypothesis

  4. Fermat's Last Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The P versus NP Problem is one of the most famous unsolved problems in Complexity Theory. It asks whether every problem in NP can be solved in polynomial time.

Multiple choice

What is the name of the theorem that states that every positive integer can be expressed as the sum of three primes?

  1. Fermat's Last Theorem

  2. Goldbach's Conjecture

  3. Catalan's Conjecture

  4. Hardy–Littlewood conjecture
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Hardy–Littlewood conjecture states that every positive integer can be expressed as the sum of three primes.

Multiple choice

Which of the following is NOT a prime number?

  1. 2

  2. 3

  3. 5

  4. 9
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

9 is a composite number because it has factors other than 1 and itself (e.g., 3).

Multiple choice

What is the smallest positive integer n such that n^2 + 1 is a prime number?

  1. 2

  2. 3

  3. 5

  4. 7
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We can check each option: 2^2 + 1 = 5 (prime), 3^2 + 1 = 10 (not prime), 5^2 + 1 = 26 (not prime), 7^2 + 1 = 50 (not prime). Therefore, n = 2.

Multiple choice

Find the number of positive integers less than 100 that are relatively prime to 100.

  1. 20

  2. 40

  3. 60

  4. 80
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

To be relatively prime to 100, a number must not share any common factors with 100. The prime factorization of 100 is 2^2 * 5^2. Therefore, any number that does not contain these factors is relatively prime to 100. There are 40 such numbers less than 100.

Multiple choice

Which of the following is a Fermat prime?

  1. 3

  2. 7

  3. 17

  4. 23
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A Fermat prime is a prime number of the form 2^(2^n) + 1. 17 is a Fermat prime because 2^(2^2) + 1 = 17.

Multiple choice

Which of the following is a Mersenne prime?

  1. 3

  2. 7

  3. 17

  4. 31
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A Mersenne prime is a prime number of the form M_p = 2^p - 1, where p is a prime number. 31 is a Mersenne prime because 2^31 - 1 = 31.

Multiple choice

In computer science, which of these problems is often solved using number theory?

  1. Primality testing

  2. Integer factorization

  3. Finding the greatest common divisor

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Primality testing, integer factorization, and finding the greatest common divisor are all problems that are frequently encountered in computer science and are often solved using techniques from number theory.

Multiple choice

In number theory, what is the name of the theorem that states that every positive integer can be written as a unique product of prime numbers?

  1. Fundamental theorem of arithmetic

  2. Goldbach's conjecture

  3. Riemann hypothesis

  4. Fermat's Last Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The fundamental theorem of arithmetic states that every positive integer can be written as a unique product of prime numbers. This theorem is a cornerstone of number theory and has many applications in various fields.

Multiple choice

In cryptography, what is the term for a large prime number that is used as part of a public key?

  1. Public exponent

  2. Private exponent

  3. Modulus

  4. Prime factor
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In public-key cryptography, the modulus is a large prime number that is used as part of the public key. The modulus is used to encrypt messages and verify digital signatures.

Multiple choice

Which of these is an example of a number-theoretic problem that is still unsolved?

  1. Goldbach's conjecture

  2. Twin prime conjecture

  3. Collatz conjecture

  4. All of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Goldbach's conjecture, twin prime conjecture, and Collatz conjecture are all examples of number-theoretic problems that have been studied extensively but remain unsolved. These problems continue to challenge mathematicians and attract significant research efforts.

Multiple choice

In number theory, what is the name of the theorem that states that for any positive integer n, there are infinitely many primes p such that p ≡ 1 (mod n)?

  1. Dirichlet's theorem on primes in arithmetic progressions

  2. Chebyshev's theorem on prime numbers

  3. Prime number theorem

  4. Twin prime conjecture
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Dirichlet's theorem on primes in arithmetic progressions states that for any positive integer n, there are infinitely many primes p such that p ≡ 1 (mod n). This theorem has important applications in number theory and its subfields.

Multiple choice

Which of the following is an example of a prime ideal in the ring of integers ℤ?

  1. The set of even integers

  2. The set of odd integers

  3. The set of prime numbers

  4. The set of composite numbers
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The set of prime numbers forms a prime ideal in the ring of integers ℤ. It is an ideal because it is non-empty, closed under addition, and closed under multiplication by an integer. It is prime because it is not contained in any larger ideal other than itself and the whole ring ℤ.