# Real Numbers

Concepts Lectures on Real Numbers and Divisibility

## Contents

## Concept Learning Sessions

Session # 1 What are Real Numbers?

Session # 2 What is meant by divisibility?

Session # 3 Properties of divisibility : Part 1

Session # 4 Properties of divisibility : Part 2

Session # 5 Euclid's Division Lemma : Part 1

Session # 6 Euclid's Division Lemma : Part 2

Session # 7 Euclid's Division Lemma : Part 3

Session # 8 Proof of the Euclid's Division Lemma

Session # 9 Euclid's Division Lemma - Applications

## Problems

Problem Solving Session # 1 Show that n^2-1 is divisible by 8 if n is an odd positive integer.

Problem Solving Session # 2 Show that the square of any positive integer is of the form of 3m or 3m+1 for some integer m.

Problem Solving Session # 3 Prove that one of every three consecutive positive integers is divisible by 3

## Concept Learning Sessions

Session # 10 Euclid's Division Algorithm

Session # 11 Euclid's Division Algorithm: What is GCD?

Session # 12 Euclid's Division Algorithm: What are co-prime numbers?

Session # 13 Euclid's Division Algorithm: Finding GCD of Two Positive Integers

Session # 14 Euclid's Division Algorithm: Theorem 1 and its proof

Session # 15 Expressing GCD of Two Positive Integers as a Linear Combination

## Problems

Problem Solving Session # 4 Find the GCD/HCF of 237 and 81 and express it as a linear combination of 237 and 81

Problem Solving Session # 5 Find the GCD/HCF of 72 and 56 and express it as a linear combination of 72 and 56. Also show that the linear combination is not unique.

Problems Solving Session # 6 Word Problem

Problems Solving Session # 7 Word Problem

## Concept Lectures

Session # 16 Fundamental Theorem of Arithmetic

## Problems

Problems Solving Session # 8 Express 168 and 234 as a product of prime factors

Problems Solving Session # 9 Prove that there is no natural number n for which 4^n ends with digit zero

Problem Solving Session # 10 Prove that there are infinitely many prime numbers

Problem Solving Session # 11Prove that a positive integer n is a prime number, if no prime less than or equal to square root of n, divides n