Euclid's Division Algorithm is a systematic method for finding the Highest Common Factor (HCF) of two positive integers. Based on Euclid's Division Lemma, it states that for any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that a = bq + r, where 0 ≤ r < b. This algorithm is a fundamental concept in the CBSE Class 10 Real Numbers chapter.
Euclid's Division Lemma states that for any two positive integers a and b, there exist unique non-negative integers q and r such that a = bq + r, where 0 ≤ r < b. Here, a is the dividend, b is the divisor, q is the quotient, and r is the remainder. This lemma forms the foundation of Euclid's Division Algorithm and is used extensively in number theory.
To find the HCF of two positive integers a and b (where a > b): (1) Apply the division lemma to a and b to get a = bq + r. (2) If r = 0, then b is the HCF. (3) If r ≠ 0, apply the division lemma to b and r. (4) Continue repeating the process until the remainder becomes 0. The divisor at the last step (when the remainder is 0) is the HCF of a and b.
The algorithm works because HCF(a, b) = HCF(b, r), where r is the remainder when a is divided by b. Each step reduces the size of the numbers involved, and since the remainders form a strictly decreasing sequence of non-negative integers, the algorithm must terminate in a finite number of steps.
Euclid's Division Algorithm is used in a wide range of mathematical applications: finding HCF of large numbers, simplifying fractions, solving Diophantine equations, and in cryptography (RSA algorithm). In CBSE exams, it frequently appears as 3-mark or 5-mark questions.
Euclid's Division Lemma is the mathematical statement a = bq + r. Euclid's Division Algorithm is the process of repeatedly applying the lemma to find the HCF of two numbers.
The standard algorithm applies to positive integers. For negative integers, take their absolute values first and then apply the algorithm.
If one number is a multiple of the other, the remainder is 0 in the first step itself, and the smaller number is the HCF.
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