Download Formula Book for Class 10 Maths and Sciece

## NCERT Solutions for Class 10th Maths: Chapter 1 Real Numbers

Exercise 1.1

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Exercise 1.2

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Exercise 1.3

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Exercise 1.4

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Q2

i ii iii iv v vi vii viii ix x

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**1. Euclid’s division lemma :**

Given positive integers a and b, there exist whole numbers q and r satisfying \(a = bq + r, 0 ≤ r < b\).

**2. 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:

Step 1 : Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b.

Step 2 : If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r.

Step 3 : 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).

**3. 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.

4. If p is a prime and p divides a2, then p divides q, where a is a positive integer.

5. Let x be a rational number whose decimal expansion terminates. Then we can express x in the form p/q , where p and q are co-prime, and the prime factorisation of q is of the form \(2^n . 5^m\), where n, m are non-negative integers.

6. Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2^n .5^m,where n, m are non-negative integers. Then x has a decimal expansion which terminates.

7. Let x = p/q be a rational number, such that the prime factorisation of q is not of the form \(2^n . 5^m\), where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).