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Key Notes, Chapter 01, Real Numbers, •, •, , •, , •, , •, , •, , For given positive integers ‘a’ and ‘b’ there exist unique whole numbers ‘q’ and ‘r’ satisfying, the relation a = bq + r , 0 ≤ r < b. ., Euclid’s division algorithms: HCF of any two positive integers a and b. With a > b is, obtained as follows:, Step 1: Apply Euclid’s division lemma to a and b to find q and r such that, a = bq + r , 0 ≤ r < b., a= Dividend, b=Divisor, q=quotient, r=remainder, Step II: If r = 0, HCF ( a, b ) = b if r ≠ 0, apply Euclid’s lemma to b and r., Step III: Continue the process till the remainder is zero. The divisor at this stage will be the, required HCF, The Fundamental Theorem of Arithmetic: Every composite number can be expressed, (factorized) as a product of primes and this factorization is unique, apart from the order in, which the prime factors occur. Ex : 24 = 2 × 2 × 2 × 3 = 3 × 2 × 2 × 2, p, Let x = , q ' ≠ 0 to be a rational number, such that the prime factorization of ‘q’ is of the, q, form 2m 5n, where m, n are non-negative integers. Then x has a decimal expansion which is, terminating., p, Let x = , q ≠ 0 be a rational number, such that the prime factorization of q is not of the, q, form 2m5n, where m, n are non-negative integers. Then x has a decimal expansion which is, non-terminating repeating., p is irrational, which p is a prime. A number is called irrational if it cannot be written in the, P, form, where p and q are integers and q ≠ 0., q