# NCERT Class 10 Mathematics Chapter 1 Real Numbers (For CBSE, ICSE, IAS, NET, NRA 2022)

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## Summary

In this chapter, you have studied the following points:

1. Euclid՚s division lemma: Given positive integers a and b, there exist whole numbers q and r satisfying .

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 , is obtained as follows:

Step 1: Apply the division lemma to find q and r where .

Step 2: If , the HCF is b. If , 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 Also, .

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 , then p divides q, where a is a positive integer.

5. To prove that are irrationals.

6. Let x be a rational number whose decimal expansion terminates. Then we can express x in the form where p and q are coprime, and the prime factorisation of q is of the form , where n, m are non-negative integers.

7. Let be a rational number, such that the prime factorisation of q is of the form , where n, m are non-negative integers. Then x has a decimal expansion which terminates.

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

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