NCERT Solutions for Class 10th Maths: Chapter 1 Real Numbers
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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 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 , 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 , where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).