# CBSE Class 10-Mathematics: Chapter – 1 Real Numbers Part 10 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 9**:

Use Euclid՚s Division Lemma to show that the square of any positive integer of the form or for some integer .

**Answer**:

Let

**Question 10**:

Prove that in is not a rational number, if n is not perfect square.

**Answer**:

Let

From (i) and (ii) is a common factor of both .

This contradicts the Assumption that are co-prime.

So, our supposition is wrong.

**Question 11**:

Prove that the difference and quotient of are irrational.

**Answer**:

Difference of

which is irrational and quotient is

**Question 12**:

Show that is divisible by , if n is an odd positive integer.

**Answer**:

Let (an odd integer)

which is divisible by

**Question 13**:

Use Euclid Division Lemma to show that cube of any positive integer is either of the form .

**Answer**:

Let

When

Hence, cubes of any positive integer is either of the form