# EXERCISE 1.3

**NCERT Solutions for Class 10 Maths Chapter 1 ( Real Numbers)**

**NCERT Solutions for Class 10 Maths Chapter 1 ( Real Numbers)**

**NCERT** Solutions for **Class 10 Maths** Chapter **1** Real Numbers of** exercise 1.3** for all boards. We have updated all the contents for the new academic session 2020-2021.

In Class IX, you began exploring the world of** real numbers** and encountered irrational numbers. We continue our discussion on real numbers in this chapter. We begin with two very important properties of positive integers in sections 1.2 and 1.3, Euclid’s division algorithm and the fundamental principle of arithmetic.

In this chapter, you have studied the following points:

- Euclid’s division lemma :

Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r,

0 ≤ r < b. - Euclid’s division algorithm: This is based on Euclid’s division lemma. According to this,

the HCF of any two positive integers a and b, with a > b, is obtained as follows:

- Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b
- If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r.
- Continue the process till the remainder is zero. The divisor at this stage will be

HCF (a, b). Also, HCF(a, b) = HCF(b, r).

- The Fundamental Theorem of Arithmetic :

Every composite number can be expressed (factorised) as a product of primes, and this

factorisation is unique, apart from the order in which the prime factors occur.

EXERCISE-1.4