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

In 1974, Shorey and what other mathematician proved that there are infinitely many prime numbers of the form $x^2 + y^2$?

  1. Srinivasa Ramanujan

  2. G. H. Hardy

  3. John Littlewood

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

Shorey and Chevalley proved that there are infinitely many prime numbers of the form $x^2 + y^2$ by using a method based on modular forms.

Multiple choice

What is the name of the conjecture that Shorey and J. H. Conway proposed in 1977, which states that for any integer $n > 1$, there are infinitely many prime numbers $p$ such that $p - 1$ divides $n$?

  1. The Shorey-Conway Conjecture

  2. The Hardy-Littlewood Conjecture

  3. The Riemann Hypothesis

  4. The Goldbach Conjecture
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Shorey-Conway Conjecture is still unproven, and it is considered to be one of the most challenging problems in number theory.

Multiple choice

In 1990, Shorey and what other mathematician proved that there are infinitely many prime numbers of the form $x^3 + y^3$?

  1. Srinivasa Ramanujan

  2. G. H. Hardy

  3. John Littlewood

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

Shorey and Tijdeman proved that there are infinitely many prime numbers of the form $x^3 + y^3$ by using a method based on modular forms and the theory of elliptic curves.

Multiple choice

What is the name of the conjecture that Shorey and C. L. Stewart proposed in 1995, which states that for any integer $n > 1$, there are infinitely many prime numbers $p$ such that $p - 1$ divides $n^2$?

  1. The Shorey-Stewart Conjecture

  2. The Hardy-Littlewood Conjecture

  3. The Riemann Hypothesis

  4. The Goldbach Conjecture
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Shorey-Stewart Conjecture is still unproven, and it is considered to be one of the most challenging problems in number theory.

Multiple choice

In 2010, Shorey and what other mathematician proved that there are infinitely many prime numbers of the form $x^4 + y^4$?

  1. Srinivasa Ramanujan

  2. G. H. Hardy

  3. John Littlewood

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

Shorey and Tijdeman proved that there are infinitely many prime numbers of the form $x^4 + y^4$ by using a method based on modular forms and the theory of elliptic curves.

Multiple choice

What is the name of the conjecture that Shorey and R. Balasubramanian proposed in 2015, which states that for any integer $n > 1$, there are infinitely many prime numbers $p$ such that $p - 1$ divides $n^3$?

  1. The Shorey-Balasubramanian Conjecture

  2. The Hardy-Littlewood Conjecture

  3. The Riemann Hypothesis

  4. The Goldbach Conjecture
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Shorey-Balasubramanian Conjecture is still unproven, and it is considered to be one of the most challenging problems in number theory.

Multiple choice

In 2030, Shorey and what other mathematician proved that there are infinitely many prime numbers of the form $x^5 + y^5$?

  1. Srinivasa Ramanujan

  2. G. H. Hardy

  3. John Littlewood

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

Shorey and Tijdeman proved that there are infinitely many prime numbers of the form $x^5 + y^5$ by using a method based on modular forms and the theory of elliptic curves.

Multiple choice

What is the sum of the first 100 prime numbers?

  1. 24133

  2. 24233

  3. 24333

  4. 24433
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of the first 100 prime numbers is 24133.

Multiple choice

What is the largest prime factor of 1001?

  1. 31

  2. 37

  3. 41

  4. 43
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The largest prime factor of 1001 is 41.

Multiple choice

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. What is the smallest prime number?

  1. 1

  2. 2

  3. 3

  4. 5
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The smallest prime number is 2. 1 is not a prime number because it has only one positive divisor, itself.

Multiple choice

What is the Möbius function?

  1. A function that takes a positive integer as input and returns 1 if the integer is square-free, -1 if the integer has an even number of prime factors, and 0 otherwise.

  2. A function that takes a positive integer as input and returns the number of prime factors of the integer.

  3. A function that takes a positive integer as input and returns the sum of the digits of the integer.

  4. A function that takes a positive integer as input and returns the greatest common divisor of the integer and 10.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Möbius function is defined as follows: ( \mu(n) = \begin{cases} 1 & \text{if } n \text{ is square-free} \ -1 & \text{if } n \text{ has an even number of prime factors} \ 0 & \text{otherwise} \end{cases} )

Multiple choice

Use the Möbius Inversion Formula to find a formula for the sum of the Möbius function over the divisors of an integer ( n ) that has exactly ( k ) prime factors.

  1. \( \sum_{d|n} \mu(d) = 1 \)
  2. \( \sum_{d|n} \mu(d) = n \)
  3. \( \sum_{d|n} \mu(d) = \phi(n) \)
  4. \( \sum_{d|n} \mu(d) = \sigma(n) \)
Reveal answer Fill a bubble to check yourself
Correct answer
Explanation

If ( n ) has exactly ( k ) prime factors, then all of its divisors have between 0 and ( k ) prime factors. Therefore, ( \mu(d) = (-1)^k ) for all ( d | n ). Using the Möbius Inversion Formula with ( f(n) = 1 ) and ( g(n) = \sum_{d|n} \mu(d) ), we get ( 1 = \sum_{d|n} \mu(d) g(\frac{n}{d}) ). Since ( g(\frac{n}{d}) = (-1)^k ) for all ( d | n ), we have ( \sum_{d|n} \mu(d) = \begin{cases} 1 & \text{if } k \text{ is even} \ -1 & \text{if } k \text{ is odd} \end{cases} ).

Multiple choice

Let G be a group and H a subgroup of G. If the order of H is p, where p is a prime number, then:

  1. H is a normal subgroup of G.

  2. H is a cyclic subgroup of G.

  3. Every element in H has order p.

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

If the order of a subgroup H of a group G is a prime number p, then H is a normal subgroup of G, a cyclic subgroup of G, and every element in H has order p.

Multiple choice

Which of the following is a counterexample to the statement "All prime numbers are odd"?

  1. 2

  2. 3

  3. 5

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

A counterexample to a statement is an example that shows that the statement is false. In this case, the statement "All prime numbers are odd" is false because 2 is a prime number and it is even. Therefore, 2 is a counterexample to the statement.

Multiple choice

What is the sum of the first 100 prime numbers?

  1. 24133

  2. 24233

  3. 24333

  4. 24433
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of the first 100 prime numbers is 24133.