INTRODUCTION:

INTRODUCTION:

Euclid’s Division Algorithm:
It is a technique to compute (divide) the HCF of two positive integers ‘a’ and ‘b’ in such a way that it leaves a remainder ‘r’ that is smaller than ‘b’.
Here a = Dividend, b = Divisor, q = Quotient and r = Remainder
Dividend = (Divisor x Quotient) + Remainder
a = b q + r where 0 ≤ r < b

Application of Fundamental Theorem of Arithmetic.
I) We use it to prove irrationality of many numbers such as √7, √11 and √13
II) We apply this theorem to explore when exactly the decimal expansion of a rational number, say \frac { p }{ q }    (q ≠ 0) is terminating and when it is non-terminating repeating.
Note: We do this by looking at the prime factorization of the denominator q of \frac { p }{ q } we can also see that the prime factorization of q will completely reveal the nature of the decimal expansion \frac { p }{ q }

 Euclid’s division lemma:
Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r, 0 ≤ r < b.

 Euclid’s division algorithm: This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers a and b, with a > b, is obtained as follows:
Step 1: Apply the division lemma to find q and r where
a = bq + r, 0 ≤r < b.
Step 2: If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r.
Step 3: Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). Also, HCF (a, b) = HCF (b, r).

The Fundamental Theorem of Arithmetic:

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

 If p is a prime and p divides a², then p divides a, where a is a positive integer

We have proved  that √2 , √3 are irrationals.

 Let x be a rational number whose decimal expansion terminates. Then we can express x in the form \frac { p }{ q } , where p and q are co prime, and the prime factorisationof q is of the form 2ⁿ 5^m, where n, m are non-negative integers.

Let x = \frac { p }{ q } be a rational number, such that the prime factorisation of q is of the form 2ⁿ 5^m  where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Let x =\frac { p }{ q }   be a rational number, such that the prime factorisation of q  is  of the form 2ⁿ 5 where n, m are non-negative integers. Then x   has a decimal expansion which is non-terminating repeating (recurring).