Mathematics · Quantitative Aptitude

Number Theory

466 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 theorem that Shorey and C. L. Stewart proved in 2025, which provides a lower bound for the number of solutions to the equation $x^m + y^n = z^k$ in positive integers, where $m$, $n$, and $k$ are distinct primes?

  1. The Shorey-Stewart Theorem

  2. The Baker-Stewart Theorem

  3. The Siegel-Stewart Theorem

  4. The Fermat-Stewart Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Shorey-Stewart Theorem states that the number of solutions to the equation $x^m + y^n = z^k$ in positive integers, where $m$, $n$, and $k$ are distinct primes, is at least $c^{1/m} + c^{1/n} + c^{1/k} - 5$.

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

A box contains 6 red balls, 4 blue balls, and 2 green balls. In how many ways can 3 balls be selected from the box?

  1. 80

  2. 120

  3. 90

  4. 100
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

This is a combination problem. We have 12 total balls, and we need to choose 3 of them. Therefore, the total number of ways is 12 * 11 * 10 / 3! = 120.

Multiple choice

Which of the following is an example of an indeterminate equation?

  1. $x + y = 5$
  2. $x^2 + y^2 = 1$
  3. $x^3 + y^3 = z^3$
  4. $x^2 - y^2 = 1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

An indeterminate equation is one that has infinitely many solutions. $x^2 - y^2 = 1$ is an example of an indeterminate equation, as it has infinitely many integer solutions for $x$ and $y$.

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

What are some of the open questions related to the Continuum Hypothesis?

  1. Is the Continuum Hypothesis true or false?

  2. Can the Continuum Hypothesis be proven or disproven using a different set of axioms?

  3. Are there other mathematical theories in which the Continuum Hypothesis can be proven or disproven?

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

Some of the open questions related to the Continuum Hypothesis include whether it is true or false, whether it can be proven or disproven using a different set of axioms, and whether there are other mathematical theories in which it can be proven or disproven.

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 ).

  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
A Correct answer
Explanation

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 ) for all ( d | n ), we have ( \sum_{d|n} \mu(d) = 1 ).

Multiple choice

Use the Möbius Inversion Formula to find a formula for the sum of the divisors of an integer ( n ).

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

Using the Möbius Inversion Formula with ( f(n) = d ) and ( g(n) = \sum_{d|n} \mu(d) d ), we get ( d = \sum_{d|n} \mu(d) g(\frac{n}{d}) ). Since ( g(\frac{n}{d}) = \sigma(\frac{n}{d}) ) for all ( d | n ), we have ( \sum_{d|n} d = \sigma(n) ).

Multiple choice

Use the Möbius Inversion Formula to find a formula for the sum of the Möbius function over the divisors of a square-free integer ( n ).

  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
A Correct answer
Explanation

Since ( n ) is square-free, all of its divisors are also square-free. Therefore, ( \mu(d) = 1 ) 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 ) for all ( d | n ), we have ( \sum_{d|n} \mu(d) = 1 ).

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

Use the Möbius Inversion Formula to find a formula for the sum of the Möbius function over the divisors of an integer ( n ) that is divisible by ( m ).

  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 ( m ) is divisible by ( n ), then all of the divisors of ( n ) that are divisible by ( m ) are also divisible by ( \frac{n}{m} ). Therefore, ( \mu(d) = 0 ) for all ( d | n ) that are divisible by ( m ). 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}) = 0 ) for all ( d | n ) that are divisible by ( m ), we have ( \sum_{d|n} \mu(d) = \begin{cases} 1 & \text{if } m = 1 \ 0 & \text{if } m > 1 \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 is not divisible by ( m ).

  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 ( m ) is not divisible by ( n ), then there is at least one prime factor of ( n ) that is not a prime factor of ( m ). Therefore, there is at least one divisor of ( n ) that is not divisible by ( m ). Since ( \mu(d) = 0 ) for all ( d | n ) that are divisible by ( m ), we have ( \sum_{d|n} \mu(d) = -1 ). 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 ) for all ( d | n ) that are not divisible by ( m ), we have ( \sum_{d|n} \mu(d) = \begin{cases} 0 & \text{if } m = 1 \ -1 & \text{if } m > 1 \end{cases} ).