#### CHAPTER AT A GLANCE

**1. Euclid’s Division Lemma: **For given any two positive integers a and b,

there exist unique integers q and r satisfying

a = bq + r, 0 :.::; r < 2.

**2. Lemma :** A lemma is a proven statement used for proving another statement.

**3. Euclid’s Division Algorithm:** Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. To get HCF of two positive integers c and d, c > d following steps are to be followed:**(I)** Apply Euclid’s division lemma to c and d to get whole numbers q and r such that c = dq + r, 0 :.::; r < d.**(ii)** If r = 0, then d is HCF of c and d. If r ‘I’- 0, apply division lemma to d and r.**(iii)** Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

**Note:****(I)** Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also.**(ii)** Euclid’s division algorithm is stated for only +ve integers but it can be extended for all integers except zero.

**4. Fundamental Theorem of Arithmetic:** Every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

The prime factorization of a natural number is unique, except for the order of its factors. In general, given a composite number x, we factorize it as

x = p1 p2 p3 ·······.Pn, where p1, p2, p3 , Pn are primes and written in

ascending order, i.e.,

p1 :-s; p2 :-s; p3:.:.:.; :.::; Pn· Ifwe combine the same primes, we will get powers

of primes.**For Example**:

So, in each of the cases prime factors of l56 is 2 x 2 x 3 x 13

Hence, we can conclude that the prime factorisation of a number is unique.

**Note:****(i)** For two positive integers a, b

HCF (a, b) x LCM (a, b)=ax b**(ii)** If p is a prime number and it divides a2, then p also, divides a where ‘a’

is the positive integer.

**5. HCF & LCM of Three Numbers**

**6. Irrational Numbers :** Number which is not a rational number whose decimal expansion is non-terminating and non-repeating

**7. Rational Numbers and Their Decimal Expansion****(I)** If denominator of a rational number is of the form 2n sm, where n, mare non-negative integers then x has decimal expansion which terminates.**(ii)** If decimal expansion of rational number terminates then its denominator has prime factorisation of the form 2n sm, where n, mare non-negative integers.**(iii)** If denominator of a rational number is not of the form 2n 5m, where n and m are non-negative integers then the rational number has decimal expansion which is non-terminating repeating

Thus we conclude that the decimal expansion of every rational number is either terminating or non-terminating repeating.

**Regarding decimal expansion of rational number x =pq integers and q not = 0, we have,** **where p, q are co-prime****(I)** x is a terminating decimal expansion if the prime factorization of q is of the form 2m 5n where m, n are non-negative integers.**(ii)** If prime factorisation of q is not of the form 2m 5n then x is a non-terminating repeating decimal expansion

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