Bézout's identity: ax+by=gcd(a,b)

Check out Max! Proof of the Division Algorithm, • Natural Numbers 1.3 Di...
Bézout's identity, ax+by=gcd(a,b),
Euclid's algorithm, zigzag division, Extended Euclidean
number theory playlist: • Playlist
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[@wobblyjelly345]

You explain this so well and the only example I have found that I can actually follow, thank you! 😊

[@halik919]

I was lazy and didn't want to watch an 18 minute video! Now, I am more than grateful I clicked and watched. Thank you, best explanation ever!

[@pablosabogal5132]

Thank you for your upbeat atittude that made me feel a lot better!

[@wajdefadool3459]

BIG Thanks man your videos helps a lot as a collage student

[@chloinger]

This was the best explanation ever! Thank you! You saved my maths exam. :)

[@khoaang5712]

Thank you for your explanation, it helps my report so much. And I really like your smile. Thank you

[@blackpenredpen]

😃

[@tejadamichelle4889]

Thank you for a very nice explaination. Keep it up 🙂

[@mzoon6823]

Thank you man, this is so useful

[@maaikevreugdemaker9210]

It feels so weird to have done calculus without having learned this stuff.. thanks!

[@lilianavalente1385]

Thank you, you explained it so well that I am confident in passing my discrete maths exam

[@stanleygondwe6859]

Excellent explaination

[@abd-elrahmanmohamed9839]

Nice as usual ! thanks!

[@NeelSandellISAWESOME]

nice, i didn't know about the lcm trick. Thanks

[@shrameesrivastav1130]

Damn, this is flipping awesome explanation! ♥

[@MathForLife]

Nice!! My favorite part is quantifiers "forall" and "exists":D

[@blackpenredpen]

Eightc yay!

[@luissalvador6711]

instaBlaster...

[@tarunsingh7243]

Amazing explanation 👌👌

[@fadirached2247]

Very helpful!!

[@doodelay]

damn this was a really clever solution, props to bezout!

[@chronicsnail6675]

this is better than the resources my uni has thanks

[@ShahriarNafiz50]

Awesome!!!

[@Magic73805]

Thank You vey much. Sir.

[@blackpenredpen]

Yay!

[@mashroom2927]

8:57 this reminds me of finding the multiplicative inverse in ciphering class

[@srpenguinbr]

Amazing!

[@psrs985]

Woah!! Wonderful explination Love from indiaa💗

[@patipateeke]

Thanks!

[@x15cyberrush9]

thank you black pen red pen

[@edslab5383]

Great!

[@kantaprasadsinha8025]

Thank u. U may arrange a video on Bezout identity and Uclid Algorithm.

[@azzteke]

EUCLID please!

[@rohitg1529]

Can you please make a video about why Euclid’s algorithm finds the GCD? We were taught how to do this in the 6th grade I think, but I never thought about why it works until now.

[@sharonjuniorchess]

This might help medium.com/i-math/why-does-the-euclidean-algorithm-work-aaf43bd3288e

[@jacksonsingh855]

Please do a videos on how to find inverse of a function

[@112BALAGE112]

If you can't solve the equation y=f(x) for x then use: en.wikipedia.org/wiki/Lagrange_inversion_theorem

[@NotYourAverageNothing]

I have a few suggestions for videos. Here’s just one: Two circles of radius R intersect each at exactly two points. Lines are drawn from each of those points to the center of one of the circles. Those lines and the inner arc of the other circle define a region. What is its maximum area?

[@NotYourAverageNothing]

Shree Ganesh I don’t know the answer with 100% certainty, but… …I think (going off of memory here) it’s (-πR^2 + R√3)/6.

[@BigDBrian]

11:45 "I will call this x naught and y naught because why not" is what you should've said! you missed a pun opportunity. I'm disappointed.

[@shacharh5470]

Cool fact: Bezout's lemma (that's how I learned it) is actually applcable in any group that's similar enough to Z. I learned a general form of the lemma in a course on group theory. Of course in group theory the notions of gcs and lcm are defined more generally in terms of subgroups and their generating sets. Interesting stuff if you're into that sort of thing.

[@alkankondo89]

That's cool! I have taken a course on group theory but have not heard of Bezout's Lemma. I'll have to look more into that.

[@1234Daan4321]

Why do x and y have to be integers? If you fill in complex numbers for m they cancel out as well, right?

[@mjones207]

I was so hoping at 11:43 for "I will call this x-naught and y-naught because... why not?"

[@blackpenredpen]

mjones207 oh... I should have done that. :)

[@jameshenner5831]

Hey Blackpenredpen, do a video on the (complex valued) infinite series Sum(i^(n-1)/n) from n = 1 to infinity.

[@halanasser5589]

How to use it if a+b is the exponent and thier gcd(a,b)=1 ??

[@dariushuang1115]

for people unfamiliar with the Euclid's Algorithm, it's actually based upon the lemma: Suppose b = aq + r, then gcd(a, b) = gcd(a, r). You can prove this lemma by contradiction in ~8 lines

[@mattgsm]

What do the pen colours represent? Is the black colour to do with the original question while red is an insertion or something?

[@PhilBoswell]

The Number Theory playlist in the description contains a private video at #4: I have watched all the others so what am I missing?

[@sahhaf1234]

Superb..

[@pamameh1841]

Don't the possible values for Y have to be in the range 0 to (n-1) in our case 0 to (432-1)? ie 7 +432 = 439 but that's greater than 432. so only one solution?

[@gdudhdydhsudjdu6350]

How can we prove the validity of a theorem with a single example?

[@yosafendrafendra7960]

Hei I really like your posts!! but i have a question for you bout 3D topic. ABCD.EFGH Cube, its side is a. Point O is intersection between AC and BD. Determine the distance between line EO and line HB..please answer thiss

[@benjaminbrady2385]

Do the numbers on the side of Euclid’s algorithm always multiply to also give you the gcd or is that a coincidence

[@elie.makdissi]

ikr i noticed that also

[@General12th]

Should I multiply it out?

[@sharpnova2]

I haven't watched this video yet but just judging from the very first statement you wrote at the beginning isn't this just the extended euclidean algorithm

[@sergiogarofoli573]

I have an issue here: After Bézout's identity, ax+by=gcd(a,b), My problem is such as "ax" is known N and "by" is the unknown from the type [-(x-1)*gcd(a,b)] and of course gcd(a,b) is the unknown I'm looking for.

[@joanlegada5370]

so the final x is -2+7m and final y is 7-24 m?? :)

[@sutgdi749]

why he holding that black ball ?

[@alandaniels2095]

Why does he have ball in his hands?

[@jiajunxu614]

proof is more difficult than application

[@riteshrastogi5388]

there is also a blue pen LOL : )

[@znarwhal4530]

What does the upside-down A and the backwards E mean?

[@benplayzgames1]

AddQ he literally said it in the video

[@vanessakitty8867]

For All and There Exists

[@SleepyRickyC92]

AddQ Quantifiers. You will encounter it in a discrete math or logic course.

[@chronicsnail6675]

whats so important about bezouts indentity

[@stanleygondwe6859]

❤❤❤

[@siggelicious4485]

fucking genius

[@chronicsnail6675]

your way of finding gcd is better than text book

[@worsethanjoerogan8061]

It's technically the same way, but I agree that writing it out in that equation style is confusing as hell

[@Abohafsah]

Change your lighting system, board can’t be seen easily

[@matteovitellaro1082]

I love you

[@RealtyTaj]

@4:41 you had a mistake the gcd(436, 126) = 2

[@enginermis4626]

wtf

[@makshudulislam7442]

We want more IMO Problem

[@sharonjuniorchess]

Question: A dealer bought a number of horses at $344.00 each, and a number of bullocks at $265.00 each. He then discovered that the horses had cost him in all $33.00 more than the bullocks. Now, what is the smallest number of each that he must have bought? [Source: 536 Puzzles by Dudeney #20]

[@maxhaibara8828]

Bezout Identity? I thought the name is Extended Euclidean

[@shacharh5470]

It's not an identity, I don't know why he calls it that. It's an existence claim so clearly not an identity. As for the name, I've learned it as "Bezout's lemma", there's a version of it for integers, a version of it for polynomials and a generalized version of it for groups

[@sugarfrosted2005]

The extended Euclidean ALGORITHM is how you find the integers x and y in Euclidean domains (such as the integers). The Bezout identity is the resulting equation. There are mathematical structures where such x and y exist, but even the normal Euclidean algorithm doesn't work. As a general rule, the result isn't the algorithm.

[@sugarfrosted2005]

Shachar H Wikipedia calls it the Bezout identity and I think I've heard that term used in Bezout domians (things where the x and y always exist).

[@eleazaralmazan4089]

Both Bezout's Identity and Bezout's Lemma are correct.

[@mihaipuiu6231]

Dear prof. Blackpenredpen...you are very good speaker,,...your problems are very interesting, but..please...your carioca is writing sometimes, hard for me to read symbols. Can you improve this symbol's visibility?

[@stevengottlieb6023]

I am sorry, but you did not show that ALL solutions are in the form x=-2+7m and y=7-24m. In theory there can be other solutions then the ones you showed.

[@blackpenredpen]

Steven Gottlieb I agree with you. But I was just showing an example on how this works. Notice the video is 18 minutes already. Max or I will work out the proof for that in the future on another video.

[@NotYourAverageNothing]

blackpenredpen Can you plz prove that Bézout Coefficients are not unique? Are there any numbers were there *_are_* unique?

[@NotYourAverageNothing]

Steven Gottlieb It does give every solution, unless m is restricted to integers.

[@stevengottlieb6023]

Thank you for responding!

[@stevengottlieb6023]

Yes, every solution was given. The problem is that was not proven. Even BPRP agrees with that.

[@kantaprasadsinha8025]

One thing is peculiar. Upto 1960 agebra books, there was no word Euclidean algorithm,a term for determining G.C. D. All are forced to swallow this coinage.Again, Bezout' s identity is renamed as Extended Euclidean Algorithm . This is cultural imperialism.as if no civilisation did not think GCD or HCF

[@OonHan]

Hi!

[@yousefalyousef59]

let: 1/(a-b)(a+b)=A/(a+b)+B/(a-b) and form it ■A(a-b)+B(a+b)=1 ■a(B+A)+b(B-A)=1 Here are two cases of a Bezout's Lemma. say some thing about that.

[@eipiwau]

You are calling it y naught because why not xD

[@manu_j_]

9:02 something magic happened

[@Anteater23]

Did my 'Calc 3' paper today. Was nervous and made a couple stupid errors and I didn't get a couple of things but on the whole it went ok. One mistake was 4r^4 x 0 x 0 = 4r^4

[@mathematicstvc6000]

Hello

[@blackloop1861]

Can you do a video about 1+2+3+4....=-1/12

[@BigDBrian]

mathologer

[@FlexThoseMuscles]

whohoo solved

[@DeViLTh0rn]

“Because we are know we are smart, don’t do that “ 😂😂😂

[@john-athancrow4169]

WHY A.L. AGAIN!!!!!!!!!!

[@blackpenredpen]

Ιωάννης - Αθανάσιος Χαραλάμπους What's AL?

[@kenichimori8533]

Ensemble theory d(^^)

[@eliascaeiro5439]

0:19 "Okay in this video I'm going to demonstrate one of the most useful facts in number theory." but you didn't prove anything, you just showed an example. Honestly I must say I'm a bit disappointed with your videos lately. Anyway, it's just what I think so don't let it bother you.

[@ansonngpersonalgoogleaccou5104]

He 'demonstrated'

[@blackpenredpen]

I did demonstrate. Usually this is easier when we first see an example then see the proof.

[@eliascaeiro5439]

Okay, then sorry for what I said. But I still prefer proofs than examples.

[@blackpenredpen]

Max does lots of proofs : ) I will do so too later.

[@eliascaeiro5439]

Yeah I know, I subscribed when you uploaded "Practice your Trig".