NCERT Class 10 Mathematics for Blind Students made Screen Readable by Professor T K Bansal.

In this chapter, you have studied the following points:

1. Euclid’s division lemma:
Given positive integers α and β, there exist whole numbers q and r satisfying α = βq + r, where, 0 ≤ r < β.

2. Euclid’s division algorithm: This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers α and β, with α > β, is obtained as follows:
Step 1: Apply the division lemma to find q and r where α = βq + r, where, 0 ≤ r < β.
Step 2: If r = 0, the HCF is β. If r ≠ 0, apply Euclid’s lemma to β and r.
Step 3: Continue the process till the remainder is zero. The divisor at this stage will be the HCF (α, β). Also, HCF (α, β) = HCF (β, r).

3. The Fundamental Theorem of Arithmetic:
Every composite number can be expressed (factorized) as a product of prime numbers, and this factorization is unique, apart from the order in which the prime factors occur.

4. If p is a prime and p divides α^2, then p divides α, where α is a positive integer.

5. To prove that √2, √3 is irrationals.

6. 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 factorization of q is of the form 2^n × 5^m, where n, m are non-negative integers.

7. Let x = p/q be a rational number (where p and q are co-prime), such that the prime factorization 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.

8. Let x = p/q be a rational number (where p and q are co-prime), such that the prime factorization 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).

A Note to the Reader

You have seen that:
HCF (p, q, r) × LCM (p, q, r) ≠ p × q × r, where p, q, r are positive integers (see Example 8). However, the following results hold good for three numbers p, q and r:

LCM (p, q, r) = p.q. r. HCF (p, q, r) /HCF (p, q). HCF (q, r). HCF( p,r) HCF (p, q, r) = p.q.r. LCM(p, q, r) / LCM(p, q).LCM(q, r).LCM(p,r)

Congratulations! You have completed this chapter. I hope you enjoyed studying this chapter. In case you found any difficulties in this chapter or have any suggestions to improve it, please write to us at ‘blind2Visionary@gmail.com’.

End of the Chapter 1. Completed on 29 May 2023.