# NCERT Class 10 Chapter 1 Real Numbers Official CBSE Board Sample Problems Short Answer (For CBSE, ICSE, IAS, NET, NRA 2022)

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## Question

Find the HCF of 392 and 732 using Euclid՚s Division lemma.

### Solution

By Euclid՚s Division Algorithm we get

## Question

Can the HCF and LCM of two numbers be 27 and 288?

### Solution

HCF is always a factor of the LCM but we observe that 27 is not a factor of 288. So these are not the I-KF and LCM of two numbers

## Question

Show that is an irrational number.

### Solution

We will do a proof by contradiction Assume is rational then it can be expressed as a ratio, a fraction

Since p and q are positive integers, p-3q is also a positive integer. let՚s call it r for simplicity.

LHS is irrational, but RHS is rational so LHS is not equal to RHS.

But it leads to a contradiction . Therefore is an irrational number.

## Question

Prove that

is irrational.

### Solution

Let be rational,

where a and b are integers which are coprime and .

and 2b are integers which implies that is a rational number.

Which implies that is also rational.

This contradicts the fact that is an irrational number, hence our assumption is wrong.

Thus is an irrational.

## Question

Show that every positive even integer is of the form 2m and that every positive odd integer is of the form, where m is some integer.

### Solution

Let ‘a’ be any positive integer and

By Euclid՚s division algorithm

(i.e..)

(where )

(where )

If a is the form of 2m then ‘a’ is an even integer and positive odd integer is of the form

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