Real Numbers
Concepts Lectures on Real Numbers and Divisibility
Concept Lectures
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 Lectures
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
