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

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a + b + c = 0$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a consequence of Fermat's Last Theorem.

Multiple choice

Let $p$ be a prime number. If $a$ and $b$ are integers such that $a^p + b^p = c^p$, then $a^2 + b^2 = c^2$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

This is false. For example, $3^2 + 4^2 = 5^2$, but $3^4 + 4^4 eq 5^4$.

Multiple choice

What is the Goldbach Conjecture?

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

  2. Every odd number greater than 1 can be expressed as the sum of three primes.

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

  4. Every number greater than 5 can be expressed as the sum of five primes.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Goldbach Conjecture states that every even number greater than 2 can be expressed as the sum of two primes.

Multiple choice

What is the name of the mathematical theorem that states that the product of two prime numbers is always an odd number?

  1. Pythagorean theorem

  2. Euler's formula

  3. Triangle inequality theorem

  4. Prime number theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The prime number theorem states that the product of two prime numbers is always an odd number.

Multiple choice

What is the largest known prime number?

  1. 2^82,589,933 - 1

  2. 2^77,232,917 - 1

  3. 2^74,207,281 - 1

  4. 2^69,725,937 - 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The largest known prime number as of 2023 is 2^82,589,933 - 1, which has 24,862,048 digits.

Multiple choice

What is the Goldbach conjecture?

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

  2. Every odd integer greater than 3 can be expressed as the sum of three primes.

  3. Every integer greater than 5 can be expressed as the sum of four primes.

  4. Every integer greater than 7 can be expressed as the sum of five primes.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

Multiple choice

I am a number that is the sum of two consecutive prime numbers. What am I?

  1. 10

  2. 12

  3. 14

  4. 16
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The only pair of consecutive prime numbers that sum to 12 is 5 and 7.

Multiple choice

I am a number that is the sum of the first three consecutive prime numbers. What am I?

  1. 17

  2. 19

  3. 21

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

The first three consecutive prime numbers are 2, 3, and 5, which sum to 17.

Multiple choice

I am a number that is the sum of the first four consecutive prime numbers. What am I?

  1. 29

  2. 31

  3. 33

  4. 35
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The first four consecutive prime numbers are 2, 3, 5, and 7, which sum to 29.

Multiple choice

I am a number that is the sum of the first five consecutive prime numbers. What am I?

  1. 41

  2. 43

  3. 47

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

The first five consecutive prime numbers are 2, 3, 5, 7, and 11, which sum to 41.

Multiple choice

What is the Goldbach conjecture?

  1. A conjecture that states that every even integer greater than 2 can be expressed as the sum of two primes.

  2. A conjecture that states that every even integer greater than 2 can be expressed as the sum of three primes.

  3. A conjecture that states that every even integer greater than 2 can be expressed as the sum of four primes.

  4. A conjecture that states that every even integer greater than 2 can be expressed as the sum of five primes.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Goldbach conjecture is a famous unsolved problem in number theory. It states that every even integer greater than 2 can be expressed as the sum of two primes. The conjecture has been verified for all even integers up to 4³10^{18}, but a general proof remains elusive.

Multiple choice

What is the Twin Prime Conjecture?

  1. A conjecture that states that there are infinitely many pairs of prime numbers that differ by 2.

  2. A conjecture that states that there are infinitely many pairs of prime numbers that differ by 3.

  3. A conjecture that states that there are infinitely many pairs of prime numbers that differ by 4.

  4. A conjecture that states that there are infinitely many pairs of prime numbers that differ by 5.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Twin Prime Conjecture is a famous unsolved problem in number theory. It states that there are infinitely many pairs of prime numbers that differ by 2. The conjecture has been verified for all prime numbers up to 10^{18}, but a general proof remains elusive.

Multiple choice

What is the Happy Number Conjecture?

  1. A conjecture that states that every positive integer eventually reaches 1 under the Happy Number sequence.

  2. A conjecture that states that every positive integer eventually reaches a repeating cycle under the Happy Number sequence.

  3. A conjecture that states that every positive integer eventually reaches a prime number under the Happy Number sequence.

  4. A conjecture that states that every positive integer eventually reaches a perfect number under the Happy Number sequence.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Happy Number Conjecture is a famous unsolved problem in number theory. It states that every positive integer eventually reaches 1 under the Happy Number sequence. The Happy Number sequence is defined as follows: for a given positive integer n, if n is even, divide it by 2, and if n is odd, square each digit of n and add the results together. Repeat this process until n reaches 1. The conjecture states that every positive integer will eventually reach 1 under this process.

Multiple choice

What is the Strong Perfect Number Conjecture?

  1. A conjecture that states that there exists a perfect number that is also prime.

  2. A conjecture that states that there exists a perfect number that is also odd.

  3. A conjecture that states that there exists a perfect number that is also a square.

  4. A conjecture that states that there exists a perfect number that is also a cube.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Strong Perfect Number Conjecture is a famous unsolved problem in number theory. It states that there exists a perfect number that is also prime. A perfect number is a positive integer that is equal to the sum of its proper divisors (the divisors of the number excluding the number itself). A prime number is a positive integer that has exactly two divisors: 1 and itself. The conjecture states that there exists a perfect number that is also prime, which would be a very rare and interesting number.

Multiple choice

What is the Riemann hypothesis?

  1. A conjecture that relates the zeros of the Riemann zeta function to the distribution of prime numbers

  2. A conjecture that relates the zeros of the Riemann zeta function to the distribution of complex numbers

  3. A conjecture that relates the zeros of the Riemann zeta function to the distribution of real numbers

  4. A conjecture that relates the zeros of the Riemann zeta function to the distribution of integers
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Riemann hypothesis is a conjecture that relates the zeros of the Riemann zeta function to the distribution of prime numbers.