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 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

................. 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.

Multiple choice maths negative numbers and integers odd and even numbers operations on integers different types of numbers

The numbers which are not multiples of $2$ are called _______.

  1. even

  2. odd

  3. prime

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

The numbers $1,3,5,7,9$ etc are called odd number which is not divisible by $2$ and any number which ends which these numbers are also called odd number. 

So, the numbers which are not multiples of $2$ are called odd.
Hence, the answer is odd.

Multiple choice mathematical modelling proof by contradiction similar triangles

The given equation $4xy-x-y=z^2$ has:

  1. three positive integer solutions

  2. one positive integer solutions

  3. two positive integer solutions

  4. no positive integer solutions
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Suppose all the solution of the given equation are positive integers
We write the equation in the equivalent form

$(4x-1)(4y-1)=4z^2+1$.

Let $p$ be a prime divisor of $4x-1$. Then

$4z^2+1\equiv 0$(mod $p$)

or

$(2z)^2\equiv -1$ (mod $p$).

On the other hand, Fermat's theorem yields

$(2z)^{p-1}\equiv 1$ (mod $p$)

hence

$(2z)^{p-1}\equiv (2z^2)^{\frac{p-1}{2}}\equiv (-1)^{\frac{p-1}{2}}\equiv 1$(mod $p$)

This implies that $p \equiv 1$ (mod $4$). It follows that all prime divisors of $4x 1$ are congruent to $1$ modulo $4,$ hence $4x 1 1$ (mod $4$), a contradiction.

Hence they are no positive integer solutions.
Multiple choice mathematical modelling proof by contradiction similar triangles

To prove "" All prime numbers are not odd."  we showed that "$2$ is even and prime"
This method is  

  1. The Direct method.

  2. The proof by contradiction.

  3. Induction method.

  4. The proof by giving counter example.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

counterexample is a special kind of example that disproves a statement or proposition.hence it disproves that all prime numbers are odd hence it is proof by counter example

$D$

Multiple choice maths hcf-lcm introduction to multiples multiples lcm

The numbers which are multiples of $2$ are called ____.

  1. odd

  2. even

  3. prime

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

The numbers $2,4,6,8,10$ are known as even number. 

The numbers which are completely divisible by $2$ are known as even numbers.
So, the numbers which are multiples of $2$ are called even numbers.

Multiple choice maths hcf-lcm introduction to multiples multiples lcm

L.C.M. of two co-prime numbers is their

  1. sum

  2. difference

  3. product

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

The two numbers which have only 1 as their common factor are called co-primes.

For example, Factors of $ 5 $  are $ 1, 5 $
Factors of $ 3 $ are $ 1, 3 $

Common factors is $ 1 $.
So they are co-prime numbers.

To find their LCM, we

then choose each prime number with the greatest power and multiply them to get the LCM.
$ => LCM = 3 \times 5 = 15 $

Hence, LCM of two co-prime numbers is their product.

Multiple choice maths hcf-lcm introduction to multiples multiples lcm

Every number is a ...... and a ........ of itself.

  1. factor, multiple

  2. prime, composite

  3. even, odd

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

Every number is a factor and a multiple of itself. 

For example, $ 10 $ has a factor $ 10 $ as well as a multiple $ 10. $

Multiple choice introduction to ratio and percentages comparing quantities maths

There are four numbers whose product is $9261000$ and each of these four numbers is formed by $3$ distinct prime numbers. The average of all the four numbers is:

  1. $61.75$
  2. $67.25$
  3. $82.33$
  4. $Data\ insufficient$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The product of the four numbers is 9261000. If we assume the numbers are equal or near each other, the cube root of 9261000 is 210. However, the question states each number is formed by 3 distinct prime numbers. This implies a specific set of numbers. Given the options, 61.75 is the only plausible arithmetic mean.

Multiple choice maths numbers and place value many forms of ten thousand standard form of numbers number names, numerals and place values

The sum of the powers of the prime factors in $108 \times 192$  is

  1. $5$
  2. $7$
  3. $8$
  4. $12$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle 108=2\times 2\times  3\times 3\times 3=2^{2}\times 3^{3}$
$\displaystyle 192=2\times2\times2\times2\times2\times2\times3$
$\displaystyle =2^{6}\times 3^{1}$
$\displaystyle 108 \times 192=2^{2}\times 3^{3}\times 2^{6}\times 3^{1}$
$\displaystyle =2^{8}\times 3^{4}$
Sum of the powers = 8 + 4 = 12