In this video, I spell out what divisibility is in great detail. I also state and prove many properties of divisibility in several lemmas. Then I recall the Well-Ordering Axiom, and I verify the Division Algorithm. After that, I motivate the Division Algorithm through the use of several examples.
Hi everyone, welcome back to my channel, I’m Dave.
In today’s episode, I explore divisibility and the Division Algorithm. The importance of divisibility flows from the following simple observation. Notice that the sum, difference, and product of any two integers is also an integer. However, the quotient of two integers may not be any integer, e.g., 8/3 is not an integer.
Divisibility (and the Division Algorithm)
What does divisibility mean?
Say integers a and b have the property: there exists an integer c such that b= a * c. Then we say that 1) a is a divisor of b, 2) a divided b, 3) b is divisible by a, 4) b is a multiple of a, or a is a factor of b. These four words describe the same property.
What are the properties of divisibility?
The divisibility relation has essential properties such as antisymmetric, multiplicative, and exponential properties. All of these properties are straightforward and easy to prove. Moreover, the symmetric closure of the divisibility relation is almost the equals relation; in other words, if a divides b and also b divides a, then a and b are the same except for a possible difference in sign.
What is the Division Algorithm formula?
The Division Algorithm is the name of a theorem that states the following. Given any nonzero positive integers a and b, there are unique positive integers q and r such that a = b * q + r. The integers q and r are named, the quotient and remainder, respectively.
How do you prove the Division Algorithm?
Most often, the proof of the Division Algorithm uses the Well-Ordering Axiom to verify the quotient and remainder’s existence. The Well-Ordering Axiom states that every nonempty subset of positive integers has a least element. Then, the uniqueness is proven using an argument along the lines of supposing; there are two quotients and remainders that satisfy the equation in the Division Algorithm, then showing that these quotients and remainders are the same.
In conclusion, I want to to turn it over to you.
Did you follow the proof of the Division Algorithm?
So, either way, let us know what you think in the comments for this video: Divisibility right now.
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