Facts that Matter- Real Numbers Class 10 Notes | EduRev
Mathematics (Maths) Class 10
Class 10 : Facts that Matter- Real Numbers Class 10 Notes | EduRev
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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. - Euclid’s Division Lemma
For any two given positive integers ‘a’ and ‘b’ there exists unique whole numbers ‘q’ and ‘r’ such that
a = bq + r, where 0 ≥ r < b
Here, a = Dividend, b = Divisor
q = Quotient, r = Remainder
i.e., Dividend = (Divisor × Quotient) + Remainder - Euclid’s Division Algorithm
Euclid’s Division Algorithm is a technique to compute the HCF of two positive integers ‘a’ and ‘b’ (a > b) using the following steps:
Step-I: Applying Euclid’s Lemma to a and b to find whole numbers ‘q’ and ‘r’ such that:
a = bq + r, 0 ≥ r < b
Step-II: If r = 0 then ‘b’ is the HCF of ‘a’ and ‘b’. If r ≠0 then apply the Euclid’s division lemma to ‘b’ and ‘r’.
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 Euclid’s Divison Algorithm for positive integers only but it can be extended for all integers except zero i.e., b ≠0.
II. When ‘a’ and ‘b’ are two positive integers such that a = bq + r, 0 ≤ 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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