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 theorem that states that every positive integer can be expressed as a sum of three prime numbers?

  1. Pythagoras' Theorem

  2. Euler's Formula

  3. Bhaskara's Formula

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

The Hardy–Littlewood conjecture is a mathematical theorem that states that every positive integer can be expressed as a sum of three prime numbers. It was proposed by the English mathematicians Godfrey Harold Hardy and John Edensor Littlewood in 1923. The Hardy–Littlewood conjecture is still unproven, but it is one of the most famous and challenging problems in number theory.

Multiple choice

Which of the following is a prime number?

  1. 12

  2. 23

  3. 36

  4. 49
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. 23 is a prime number because it has no positive divisors other than 1 and 23.

Multiple choice

What is the fundamental theorem of arithmetic?

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

  2. Every integer greater than 1 can be expressed as a sum of prime numbers.

  3. Every integer greater than 1 can be expressed as a difference of prime numbers.

  4. Every integer greater than 1 can be expressed as a quotient of prime numbers.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The fundamental theorem of arithmetic states that every integer greater than 1 can be expressed as a product of prime numbers. For example, 12 can be expressed as 2 x 2 x 3, and 23 is a prime number.

Multiple choice

What is the Goldbach conjecture?

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

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

  3. Every integer greater than 1 can be expressed as the sum of two prime numbers.

  4. Every integer greater than 2 can be expressed as the sum of two prime numbers.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. For example, 4 can be expressed as 2 + 2, and 6 can be expressed as 3 + 3.

Multiple choice

Which of the following numbers is prime?

  1. 12

  2. 17

  3. 21

  4. 29
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. Among the given options, only 17 satisfies this condition. 12 is divisible by 2 and 3, 21 is divisible by 3 and 7, and 29 is prime.

Multiple choice

What is the prime factorization of 100?

  1. 2^2 * 5^2

  2. 2^3 * 5^2

  3. 2^4 * 5

  4. 2^5 * 5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Prime factorization involves expressing a number as a product of prime numbers. For 100, we can write it as 100 = 2 * 2 * 5 * 5 = 2^2 * 5^2.

Multiple choice

What is the sum of all prime numbers less than 20?

  1. 28

  2. 41

  3. 53

  4. 71
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Adding these numbers together, we get 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 71.

Multiple choice

Which of the following statements is true about Goldbach's conjecture?

  1. Every even integer greater than 2 can be expressed as the sum of two primes.

  2. Every odd integer greater than 5 can be expressed as the sum of three primes.

  3. Every integer greater than 1 can be expressed as the sum of four primes.

  4. Every integer greater than 10 can be expressed as the sum of five primes.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. It is one of the most famous unsolved problems in mathematics.

Multiple choice

Which of the following is a Mersenne prime?

  1. M3

  2. M5

  3. M7

  4. M11
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A Mersenne prime is a prime number of the form M_p = 2^p - 1, where p is a prime number. Among the given options, only M7 = 2^7 - 1 = 127 is a prime number. Therefore, M7 is a Mersenne prime.

Multiple choice

What is the value of the sum of the first 100 prime numbers?

  1. 24133

  2. 25501

  3. 26989

  4. 28549
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find the sum of the first 100 prime numbers, we can use a loop or a pre-calculated list of prime numbers. Summing the first 100 prime numbers, we get 2 + 3 + 5 + 7 + 11 + ... + 541 = 24133.

Multiple choice

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

  1. 2

  2. 3

  3. 4

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

We can check each of the options to see if it satisfies the condition. For n = 2, we have n^2 + n + 1 = 2^2 + 2 + 1 = 7, which is a prime number. For n = 3, we have n^2 + n + 1 = 3^2 + 3 + 1 = 13, which is also a prime number. For n = 4, we have n^2 + n + 1 = 4^2 + 4 + 1 = 21, which is not a prime number. For n = 5, we have n^2 + n + 1 = 5^2 + 5 + 1 = 31, which is a prime number. Therefore, the smallest positive integer n such that n^2 + n + 1 is a prime number is 2.

Multiple choice

What is the sum of the first 100 prime numbers?

  1. 22995

  2. 24133

  3. 25301

  4. 26497
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The first 100 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ..., 541. We can use a calculator to find the sum of these numbers, which is 22995.

Multiple choice

What is the largest prime factor of the number 1001?

  1. 11

  2. 13

  3. 17

  4. 19
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We can factor 1001 as follows: 1001 = 7 × 11 × 13. Therefore, the largest prime factor of 1001 is 19.

Multiple choice

What is the name of the puzzle in the Sadratnamala that involves finding the sum of the first 100 prime numbers?

  1. The Prime Sum Problem

  2. The Prime Number Problem

  3. The Mersenne Prime Problem

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

The puzzle in the Sadratnamala that involves finding the sum of the first 100 prime numbers is called the Prime Sum Problem.

Multiple choice

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

  1. Goldbach's conjecture

  2. Hardy-Littlewood conjecture

  3. Erdős-Straus conjecture

  4. Bombieri-Vinogradov theorem
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The Hardy-Littlewood conjecture states that every positive integer can be expressed as the sum of three prime numbers. It was first proposed by G. H. Hardy and J. E. Littlewood in 1923 and remains unproven.