NCERT Solutions for Chapter: Real Numbers, Exercise 1: Exercise

The prime factorisation of a prime number is the number itself. How many factors and prime factors does the square of a prime number have?

$\left(n^2+3n-4\right)$ can be expressed as a product of only $2$ prime factors where $n$ is a natural number. Find the value(s) of $n$ for which the given expression is an even composite number. Show your work and give valid reasons.

$\sqrt{n}$ is a natural number such that $n>1$. Which of these can definitely be expressed as a product of primes.

$i) \sqrt{n}$

$ii)\hspace{0.17em}n$

$iii)\hspace{0.17em}\frac{\sqrt{n}}{2}$

Let $p$ be a prime number and $k$ be a positive integer. If $p$ divides $k^2$, then which of these is definitely divisible by $p$?

Prove that $2\sqrt{3}+\sqrt{5}$ is irrational.

Which number is between $\sqrt{50}$ and $\frac{5\mathrm{\pi }}{2}$?

Two statements are given below - one labelled Assertion (A) and the other labelled Reason (R). Read the statements carefully and choose the option that correctly describes statements (A) and (R).
Assertion (A): $2$ is a prime number.
Reason (R): The square of an irrational number is always a prime number.

In a seminar, the number of participants in Physics, Chemistry and Mathematics are $96, 36,$ and $180$ respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.