Maths (Real Numbers)

In this chapter we will begin with a brief recall of divisibility on integers and will state some important properties of integers namely, Euclid's Division Lemma, Euclid's Division Algorithm and the fundamental theorem of arithmetic.

Euclid’s Division Lemma

It tells us about divisibility of integers. It states that any positive integers 'a' can ve divided by any other positive integers 'b' is such a way that it leaves a remainder 'r' that is smaller than 'b'. This is nothing bt a usual long division process.

Euclid’s Algorithm

Euclid's Division lemma provides us step-wise process to compute HCF of two positive integers. This step-wise procedure is known as Euclid's Algorithm.

Divisibility

• A non-zero integer ‘a’ is said to divide an integer ‘b’ if there exists an integer ‘c’ such that (b=ac)

• The integer ‘b’ is called the dividend, integer ‘a’ is known as the divisor and the integer ‘c’ is known as the quotient.

(Theorem 1) :- Fundametal Theorem of Arithmetic

Every integer > 1 is either a prime or can be uniquely factored as a product of prime numbers (ignoring the order of factors).

Properties of Divisibility

• +-1 divides every non-zero integer i.e, +-1|a for every non-zero integer ‘a’.

• 0 is divisible by every non-zero integer ‘a’ i.e a|0 for every non-zero integer ‘a’.

• 0 does not divide any integer.

• if ‘a’ is a non-zero integer and ‘b’ is any integer, then a|b => a| -b, -a| b, and -a|-b.

• if ‘a’ and ‘b’ are non-zero integers, then a|b and b|a => a=+-b.

THEOREM 2

Let ‘p’ be a prime number and ‘a’ be a positive integer. If ‘p’ divides a**2, then ‘p’ divides ‘a’.

Relation between HCF and LCM of two positive integers.

HCF X LCM = a x b

• [NOTE] :- The product of two positive integers is equal to the product of their HCF and LCM, but the same is not for three or more positive integers.