Discrete Math - 4.1.1 Divisibility

The definition and properties of divisibility with proofs of several properties. Formulas for quotient and remainder, leading into modular arithmetic.

Video Chapters:
Introduction 0:00
Division 0:04
Properties of Divisibility 1:50
Division Algorithm 9:39
Quotient and Remainders 12:19
Up Next 17:23

Textbook: Rosen, Discrete Mathematics and Its Applications, 7e

[bleghssedbe]

I'm entering Week 3 of 8 with Discrete Mathematics and Linear Algebra online, and I just wanted to say thank you so much for making your videos available publicly. They have been a critical tool in helping me learn and understand the material. Next term Statistics, and I'll be back to utilize the videos you have for that course too!

[Puddin_]

Thank you for providing the explanations, which are far more understandable than reading them all in a book.

[ymperformance]

Thank you so much! I am in an online Discrete math class with no lecture! This resource has been essential to me understanding these concepts! Thanks!

[nullrunner]

My professor is not very good at teaching these concepts, so this video is a lifesaver.

[mr.anonymous]

Thank you so much! My professor just reads off the slides with a very thick accent. I don't understand anything that is going on in the class; however, your videos are quite helpful.

[aymenechchalim]

I have been looking for such videos for a very long time, worth calling this a treasure !
can't thank you enough

[stargate]

I love you soooo much Kimberley. I have a final coming very soon, and you have been very helpful. Besides, I like your pedagogical methods. Thank you!!!

[maedre]

There is a slight error in the original definition. a|b iff ∃c: ac = b (a, b ∈ Z, c ∈ Z+). c just has to be an element of Z, not Z+ (confirmed with the book). You use that fact later in the video for 10 | -112. Just wanted to point that out in case someone like me got confused with that portion of the video.

[seanc]

thank you very very much for very efficient and effective teaching videos. the way you teach empowers me so much.

[romancampbell]

Is there a secret to understanding proofs, I can do the math but these proofs absolutely kill me.

[keldonchase]

Hi Kimberly:
I had a question regarding whether or not the following proof is legitimate.
Prove that for all integers {a, b, c}, if a|b and b|c, then a|c.

Let {a, b, c, k, j, m} all be integers.

GIVENS: a|b ak = b a = b/k

GIVENS: b|c bj = c

PROVE: a|c am=c m = c/a

To prove that a|c, we would have to show that m is an integer.

We have to show that (c/a) simplifies to be an integer.

c/a

bj / (b/k)

Copy Dot Flip

bj * (k/b)

j*k

Integers Closed Under Multiplication

We find that m was an integer.

Thus, it is true that a|c.


Is this a legitimate proof or is there a flaw in the argument?

Thank you so much!!

[ProPatriaMex]

Thank you a lot. Helped me solve my assignment!

[techjesus]

while studying this book some things feel very basic and I feel like i already know all this, but when I challenge myself with proofs or questions at the end of the section, I find some of them hard. I tend to get frustrated because sometimes the questions look very basic but i just don't know how to prove them. Is this normal or should i pursue something other than computer science as a career?

[GarryBurgess]

For some reason I was not subscribed. I fixed that.

[sagivalia]

Hey there, is it mathematically correct to write a property's defintion using propositional logic? ex. (a | b) ^ (a | c) -> a | (b+c)

[gabaromar]

division by zero is undefinde 14|0=? shoud we write q and r = 0

[femaledeer]

I am not sure how b +c =a (s +t) translates to therefore a | (b+c)

[JohnWickXD]

But isn't it confusing...first it is said that a|b iff b/a € Z...
Then we come to quotient and reminders because division isn't always even.
Doesn't this conflict with the original defination?
Like first we say in def that a|b only iff c is an integer... then we are okay with float values.😅
@Kimberly Brehm

[pcjee]

I have one question here, from the definition of divisibility, is it true that a divides b if there is an unique integer c OR for some integer c such that ac=b?

[masoudshairzadeh]

8:55 is funny for some "ASS" integer lol.