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** Facts that Matter**

**Algorithm**

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.**Lemma**

A lemma is a proven statement used for proving another statement.**E****uclidâ€™s Division Lemma**

For any two given positive integers â€˜aâ€™ and â€˜bâ€™ there exists unique whole numbers â€˜qâ€™ and â€˜râ€™ such that

a = bq + r, where 0 â‰¥ r < b

Here, a = Dividend, b = Divisor

q = Quotient, r = Remainder

i.e., Dividend = (Divisor Ã— Quotient) + Remainder**Euclidâ€™s Division Algorithm**

Euclidâ€™s Division Algorithm is a technique to compute the HCF of two positive integers â€˜aâ€™ and â€˜bâ€™ (a > b) using the following steps:**Step-I:**Applying Euclidâ€™s Lemma to a and b to find whole numbers â€˜qâ€™ and â€˜râ€™ such that:

a = bq + r, 0 â‰¥ r < b**Step-II:**If r = 0 then â€˜bâ€™ is the HCF of â€˜aâ€™ and â€˜bâ€™. If r â‰ 0 then apply the Euclidâ€™s division lemma to â€˜bâ€™ and â€˜râ€™.**Step-III:**Continue the process till the remainder is zero, i.e., repeat the step-II again and again until r = 0. Then, the divisor at this stage will be the required H.C.F.

Note:I. We state Euclidâ€™s Divison Algorithm for positive integers only but it can be extended for all integers except zero i.e., b â‰ 0.

II. When â€˜aâ€™ and â€˜bâ€™ are two positive integers such that a = bq + r, 0 â‰¤ r < b then HCF (a, b) = HCF (b, r).

**Fundamental Theorem of Arithmetic**

Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

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