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

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 surface area and volume of cube and cuboid finding out the diagonal of cube and cuboid length of the diagonal of cube diagonal of cube and cuboid

Find the pythagorean triplet.

  1. $8, 15, 17$
  2. $9, 10, 15$
  3. $9, 10, 17$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Every right triangle has side length satisfying:

a$^{2}$ $+$ b$^{2}$ $=$ c$^{2}$
c is the longest side,
Here 
$8^{2}$ $+$ $15^{2}$ $=$ $17^{2}$
Hence it is a pythagorean triplet.
Option A is correct.

Multiple choice maths surface area and volume of cube and cuboid finding out the diagonal of cube and cuboid length of the diagonal of cube diagonal of cube and cuboid

A Pythagorean triplet whose smallest member is $8$, is:

  1. $8, 15, 18$
  2. $8, 13, 16$
  3. $8, 14, 17$
  4. $8, 15, 17$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We can get Pythagorean triplet by using general form $2m,\ m^{2}-1,\ m^{2}+1 $
Let us first take 

$m^{2}-1=8$
So, $m^{2}=8+1=9$
Which gives $m=3$
Therefore $2m=6$ and  $ \displaystyle m^{2}+1 = 10  $
The triplet is thus $6,8,10$, but $8$ is not the smallest member of this triplet.

So let us try
$2m=8$
then $m=4$
We get $ \displaystyle m^{2}+1 = 16-1=15$
and $ \displaystyle m^{2}+1 =16+1=17$
The triplet is $8,15,17$ with $8$ as the smallest member.

Hence, option $D.$

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

The sides of a quadrilateral are all positive integers and three of them are $5, 10, 20.$ How many possible value are there for the fourth side?

  1. $29$
  2. $31$
  3. $32$
  4. $34$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For any quadrilateral with sides a, b, c, and d, the triangle inequality requires that the longest side is less than the sum of the other three. Here, three sides are 5, 10, and 20. If x is the fourth side, cases depend on whether x is the longest side or not, yielding possible integer values calculated via the inequality bounds.