⇒ 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.
⇒ Lemma is a proven statement used for proving another statement.
⇒ 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 and is obtained by the following method:
Step 1. Obtain two integers q and r, such that a = bq + r, 0 ≤ r < b.
Step 2. If r = 0, then b is the required HCF.
Step 3. If r ≠ 0, 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).
A number which cannot be expressed in the form of p/q, q ≠ 0 and p, q ∈ I, will be an irrational number. The set of irrational numbers is generally denoted by S.
⇒ The rational number p/q will have a terminating decimal representation only, if in standard form, the prime factorisation of q, the denominator is of the form 2n 5m, where n, m are some non-negative integers.
⇒ The rational number p/q will have a non-terminating repeating (recurring) decimal representation, if in standard form, the prime factorisation of q, the denominator is not of the form 2n 5m, where n, m are some non-negative integers.
⇒ The decimal expansion of every rational number is either terminating or non-terminating repeating.
⇒ If p is a prime and p divides a2, then p divides a, where a is a positive integer.
⇒ For any two positive integers p and q, we have
HCF (p,q) x LCM [p, q] = p x q
⇒ For any three positive integers a, b and c, we have