GC1 The Division Algorithm for ordinary integers

When you divide an integer m by another positive integer n, then you get a quotient and a remainder. As long as the remainder is non-negative and less than n, the quotient and the remainder will be unique. This is called "The Division Algorithm" and will be proved rigorously using the Well Ordering Principle. As an example of its use, it is proved that if you square an odd integer and find its remainder when divided by 8, you will always get 1.The Division Algorithm is used in Elementary #numbertheory Elementary #grouptheory and in #ringtheory Subscribe @Shahriari for more undergraduate math videos. 00:00 Introduction 00:10 What does the Division Algorithm say? A toy example. 02:38 Where is the "algorithm" and why does it work? 04:26 The Division algorithm stated mathematically 04:52 The mod notation; a = b mod n 05:44 Proof that a = b mod n if and only if n divides a-b 07:40 Application: What happens if you square an integer and find its remainder when divided by 8? 08:30 Proof of the pattern using the Division Algorithm 10:34 Proposition: If you divide the square of an integer by 8, you will never get remainder 2, 3, 5, 6, or 7 11:03 How to monetize the Proposition? 11:53 Proposition: n^2 = 1 mod 8 if n is an odd integer 12:27 Discussion: Should we prove the Division Algorithm rigorously? 13:31 The Well Ordering Principle 14:51 Proof of the Division Algorithm 17:59 Proof of Uniqueness One of a series of lectures by Shahriar Shahriari on basic mathematical concepts used in undergraduate college mathematics. Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award.