Introduction:

Euclid’s Division Lemma/Euclid’s Division Algorithm: 

Given positive integers a and b, there exist unique integers q and r satisfying a=bq+r, 0 r<b.

This statement is nothing but a restatement of the long division process in which q is called the quotient and r is called the remainder. 

NOTE:

1. Lemma is a proven statement used for proving another statement.

2. Euclid’s Division Algorithm can be extended for all integers, except zero i.e., b 0.

HCF of two positive integers : 

HCF of two positive integers a and b is the largest integer (say d ) that divides both a and b(a>b) and is obtained by the following method :

Step 1. Obtain two integers r and q, such that a=bq+r, 0r<b.

Step 2. If r=0, then b is the required HCF.

Step 3. If r0, then again obtain two integers using Euclid’s Division Lemma and continue till the remainder becomes zero. The divisor when remainder becomes zero, is the required HCF.

The Fundamental Theorem of Arithmetic :

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

Irrational Number :

A number is an irrational if and only if, its decimal representation is non-terminating and non-repeating (non-recurring).

OR

A number which cannot be expressed in the form of pq , q 0 and p, qI, will be an irrational number. The set of irrational numbers is generally denoted by Q.

NOTE:

1. The rational number pq will have a terminating decimal representation only, if in standard form, the prime factorisation of q, the denominator is of the form 2ⁿ5^m, where n, m are some non-negative integers.