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# Facts that Matter- Real Numbers Class 10 Notes | EduRev

## Class 10 : Facts that Matter- Real Numbers Class 10 Notes | EduRev

The document Facts that Matter- Real Numbers Class 10 Notes | EduRev is a part of the Class 10 Course Mathematics (Maths) Class 10.
All you need of Class 10 at this link: Class 10

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.
• Euclidâ€™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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## Mathematics (Maths) Class 10

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