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

If M = 77 $\times$ 144 $\times$ 45, which of the following groups gives the prime factors of M?

  1. 2, 3, 5, 7, 11, 13

  2. 2, 3, 5, 19

  3. 2, 3, 5, 7, 11, 13, 14

  4. 2, 3, 5, 7, 11
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

M = 77 $\times$ 144 $\times$ 45 = 7 $\times$ 11 $\times$ 12 $\times$ 12 $\times$ 9 $\times$ 5 = 7 $\times$ 11 $\times$ 2 $\times$ 2 $\times$ 3 $\times$ 2 $\times$ 2 $\times$ 3 $\times$ 3 $\times$ 3 $\times$ 5 So, prime factors of M are 2, 3, 5, 7 and 11.

Multiple choice

If M = 44 $\times$ 65 $\times$ 84, which of the following groups gives the prime factors of M?

  1. 2, 3, 5, 7, 11, 17

  2. 2, 3, 5, 7, 11, 13

  3. 3, 4, 5, 7, 11, 13

  4. 2, 3, 5, 7, 11
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

M = 44 $\times$ 65 $\times$ 84 M = 4 $\times$ 11 $\times$ 13 $\times$ 5 $\times$ 12 $\times$ 7 = 2 $\times$ 2 $\times$ 11 $\times$ 13 $\times$ 5 $\times$ 2 $\times$ 2 $\times$ 3 $\times$ 7 So, the prime factors of M are 2, 3, 5, 7, 11 and 13.

Multiple choice

If M = 34 $\times$ 65 $\times$ 133, which of the following groups gives the representation of M as a product of prime factors?
  1. 2 $\times$ 5 $\times$ 11 $\times$ 13 $\times$ 17
  2. 2 $\times$ 5 $\times$ 7 $\times$ 13 $\times$ 19
  3. 2 $\times$ 7 $\times$ 13 $\times$ 17 $\times$ 19
  4. 2 $\times$ 5 $\times$ 7 $\times$ 13 $\times$ 17 $\times$ 19
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

M = 34 $\times$ 65 $\times$ 133 M = 2 $\times$ 17 $\times$ 13 $\times$ 5 $\times$ 19 $\times$ 7     = 2 $\times$ 5 $\times$ 7 $\times$ 13 $\times$ 17 $\times$ 19

Multiple choice

If M = 33 $\times$ 221 $\times$ 203, which of the following groups gives the representation of M as a product of prime factors?
  1. 3 $\times$ 9 $\times$ 11 $\times$ 13 $\times$ 17 $\times$ 29
  2. 3 $\times$ 7 $\times$ 11 $\times$ 13 $\times$ 17 $\times$ 29
  3. 3 $\times$ 9 $\times$ 11 $\times$ 13 $\times$ 17 $\times$ 27
  4. 3 $\times$ 7 $\times$ 11 $\times$ 13 $\times$ 19 $\times$ 29
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

M = 33 $\times$ 221 $\times$ 203 M = 3 $\times$ 11 $\times$ 13 $\times$ 17 $\times$ 29 $\times$ 7 The product of prime factors of M = 3 $\times$ 7 $\times$ 11 $\times$ 13 $\times$ 17 $\times$ 29

Multiple choice maths numbers and place value forming numbers formation of greatest and smallest numbers identifying the largest and smallest numbers with given digits

Which among the following  is a four digit prime number using the digits $1,7,0,9$?

  1. $1790$
  2. $1709$
  3. $9710$
  4. $7910$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
A prime number is a number which is only divisible by itself and $1$.

(a) $1790$ is divisible by $2$, so it is not a four digit prime number.

(b) $1709$ is only divisible by itself, so it is a four digit prime number.

(c) $9710$ is divisible by $2$, so it is not a four digit prime number.

(d) $7910$ is divisible by $2$, so it is not a four digit prime number.

Hence, $1709$ is a four digit prime number using the digits $1,7,0,9$.
Multiple choice validating statements proofs in mathematics mathematical reasoning maths

Name the technique used in the solution of the problems below :

Question: Show that the following statement is false: If n is an odd integer, then n is prime.

Solution: The given statement is in the form “if p then q” we have to show that this is false, If p then ~q.


If n= 99 is odd integer which is not a prime number. Thus, we conclude that the given statement is false.

  1. Counter example

  2. Contrapositive method

  3. Direct method

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

A counterexample is a specific case that proves a general statement false. Here, n = 99 is an odd integer that is not prime, disproving the claim.

Multiple choice maths real number fundemental theorem of arithmetic real numbers on number line fundamental theorem of arithmetic


$20$ is written as the product of primes as :

  1. ${2\times 5 }$
  2. ${2\times 2\times 3\times 5}$
  3. ${2\times 2\times 5}$
  4. ${2\times 2\times 3}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To write a number as product of its primes, we divide it by various prime numbers $ 2, 3, 5, 7 $ etc one by one and check by which prime numbers it is divisible with and how many times.

Hence, $ 20 = 2 \times 10 = 2 \times 2 \times 5 $          

Multiple choice maths real number fundemental theorem of arithmetic real numbers on number line fundamental theorem of arithmetic

The ........... when multiplied always give a new unique natural number.

  1. decimal numbers

  2. fractions

  3. irrational numbers

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

For example: $24$ is made by multiplying the prime numbers $2, 2, 2$ and $3$ together. $24 = 2 \times 2 \times 2 \times 3$
It makes a unique number using a unique combination of $2, 2, 2$ and $3.$
Therefore, $D$ is the correct answer.

Multiple choice maths real number fundemental theorem of arithmetic real numbers on number line fundamental theorem of arithmetic

................. states the possibility of the prime factorization of any natural number is unique. The numbers can be multiplied in any order.

  1. Pythagora's theorem

  2. Remainder theorem

  3. Fundamental theorem of arithmetic

  4. none of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Fundamental theorem of arithmetic says that composite number can be factorised as a product of prime numbers.
Therefore, $C$ is the correct answer.