Solving Linear Congruences, Modular Arithmetic

[@alex3l4]

2:30 Correction: the gcd of a prime and another integer isn't always 1. For example if the gcd of a prime and its multiple, like gcd(7, 14) = 7.

[@tombardier]

Great explanation. I've been banging my head against the desk trying to get some of this stuff in to my head. This helped a lot, and approached it from quite a different perspective than my study material.

[@srijanadh2303]

Fr Why can i imagine you banging your head ?lmao I can really imagine it 😂

[@ibrahimiqbal8388]

The way you explained this topic needs to be appreciated. I was going through a difficult time to understand this but Thank You for making this topic easy for me.

[@personalacc4219]

watching this after 3 years of publishing it, you explain better than my dctrs thankssss for making it easy😍

[@jamesmuking]

My slides for class were horrible. This saved me from my brain fart possibly going into a brain diarrhoea into a brain dehydration

[@SequinBrain]

Excellent work detailing the specific steps in the process and not skipping any. TI-89 also has seq( from pushing 2nd > Math > List. Parameters seem to be the same.

[@subramaniabharathis]

Very good explanation. Thank you, sir. Need some more examples of difficult Sums on congruence

[@angeekevine]

Till now we are still appreciating your work Really helps

[@firanam7316]

Is there any good manual method to find the needed number at 4:06? Due to the nature of my course I am unable to use electronic supplements such as calculators, and while the method "works", it's painfully slow for larger numbers, for instance I have the congruence 61x≡205 (mod 788) - with digital methods I found that the number I'd need would be -20*788+205 = -15555 - easy to obtain digitally but rather time-consuming manually.

[@oliviacruz3839]

Thank you for explaining in detail how to solve linear congruences.

[@aborne]

You’re very welcome. These are weird, don’t you think?

[@calebpresto8684]

You can't just divide both sides of a congruence by any integer (even though you can multiply by any integer. You have to first see if gcd(n, m)=1. Only then can you divide by n.

[@SuryaBoddu]

At 7:15 what is that parametric equation? How do we get it?

[@sanjanaattanayake4231]

Thank you very much! Nice teaching with clear explanations 💙

[@Bedoroski]

Nicely presented. Thank you

[@what-it-izz]

at 2:09, for no solution, is there a reason 2 must divide 51? Where does this conclusion derive from?

[@axeldaniels9677]

Thanks, man. You really saved my ass on the final!

[@aborne]

You're welcome. Seems like this particular video does a lot of that. 🙂

[@leviticus3]

Thank you from the bottom of my heart. You are amazing. Excellent explanation!!

[@aborne]

Comments like this keep me going. I really appreciate your kind words. Cheers, -Andy

[@berat5532]

best explanation of congruences

[@a.c.4619]

So how do you take it a step further by getting the actual solution(s) for "x" using the Euclidean Algorithm? Some sites use the variables "s" and "t". I'm just looking for an easier explanation.

[@aborne]

Hi. This was as easy an explanation as I could do.

[@a.c.4619]

@@aborne I apologize for not making myself clear. I didn’t mean that I wanted an explanation simpler than the one you provided. Your explanation was excellent and easy to understand. It’s just that my textbook asks for us to delve further. Your explanation stops at simply trading one congruence modulo for another. In other words, your explanation is simply taking a congruence modulo in the form of [ax ≡ b(mod m)] into another congruence modulo of the same form only with different variables [cx ≡ d(mod m)] Again, your explanation was great. My textbook, however, does not want answers in the form of another congruence modulo. It asks for “solutions” to the original congruence modulo by utilizing the Euclidian Algorithm. The book wants it written as an equality rather than a congruence. For example, my book has the following problem: 20x ≡ 14(mod 63) I’m fairly certain that I could use your explanation to derive another congruence modulo (like you did) as follows: x ≡ 7 (mod 63) However, the textbook shows the answer written as an equality as follows: “x = -308 is a solution.” When I stated that I was searching for an easier explanation, I meant an easier explanation than my textbook and an easier explanation than other sites I’ve searched. Sorry for the ambiguity.

[@michaeljacobdelossantos]

5:44 At this part, can you do it without dividing the modulo?

[@aborne]

No, somehow you need 9x to become just x.

[@bugganaumadevi941]

Gave full clarity about solutions

[@cinderbane1]

Is the rule about GCD between at least one prime number always true? What about GCD(24,3) as an example? Isn't that = 3?

[@davidkam366]

At 1:06 , i dont think{...-27,-19,-11,-3,5,13,21,29,36...} it is the least residues system modulo 8， because they have the same remainder 5. Are u telling wrong?

[@hs1113]

Suppose that a and b are integers such that a ≡ 34 (mod 83) and b ≡ 21 (mod 83). Find an integer c such that 0 ≤ c < 83 such that 47c ≡ (53a−2 + b5)(mod 83)

[@aborne]

Good question. Perhaps post this question on r/mathquestions on reddit.

[@Ligaya_Reyes]

Thank you so much! i understand it a lot

[@Damsengeedana]

Great explanation thank you❤

[@samsunnahar9175]

Thanks a lot for excellent explanation!!

[@aborne]

You’re welcome. Have a good one!

[@Chris-ng9zi]

I got lost at the stage where you introduced the parametric equation. I know that a "congruent to b (mod n") means a= n.k +b. So b=a - n.k and so I am confused about b= 2t + 0.

[@sebastianm5047]

Amazing -- clear and concise.

[@farouqalsalih619]

Isn’t there a rule you have to follow when multiplying or dividing a number to the congruence, like it has to be coprime to the modulo number?

[@aborne]

Actually I'm not sure.

[@imranif3899]

@Andrew Borne Awesome as always. Just a tiny question for 7:15 , how did you come up with x congruent to b (mod 6)? why isn't it (mod 2) Thanks in advance!

[@janslittleclassroom6659]

Same question. How did you come up with (mod 6) instead of (mod 2)

[@charlenegwin2162]

@@janslittleclassroom6659 I don't know why, but at 7:29 he says that the solutions have to be in terms of the original mod => 9x≡ 42 (mod 6).

[@hvcreator2380]

Probably a editing mistake

[@pinkPIGSandCUPCAKES]

Thanks so much, this made it so easy to understand

[@prudhvik5603]

great work sir. Appreciated.

[@nenani1817]

WOW, I loved th eplaning of the video.

[@MrKrabs-xf2tr]

On the first problem of the one solution set, why must you look down to 6 and not 20? Is there a reason or do they all end up as the same value.

[@aborne]

Yes, they do end up the same value. The idea is to finish with the smallest number.

[@saradekeijzer8457]

Really helps, finally understand it!

[@autiskibidi]

amazing explanation tysm

[@jblkpnthr]

You are amazing!!! Thank you for this!

[@paulmacdonald5195]

Thank you for these videos.

[@ayushkumar2310]

great work!!!!!!!!!!!!!

[@PMe-my1td]

Thanks now we just need it for large numbers! e.g. 125452x - 4 = 4 mod 15044

[@aborne]

Oh my, that's a good problem for...someone else.

[@ambilyfrancis5879]

Really helpful. Thank you very much.

[@nimnanilwala5245]

Thank you sir. Great work....!

[@quetzaltpa4450]

wow, if I do not know the mod? and only know a and b? what I have to do?

[@aborne]

I mean no offense; if you are asking these questions you are studying mathematics that is a little too advanced for you at this time.

[@quetzaltpa4450]

@@aborne there is not offense.. I just want to learn..thats all

[@aborne]

@@quetzaltpa4450 Mod is short for modulo. It is sometimes represented by the this symbol, %. It is the remainder of division, for example #1, 5 mod 2 means 5 ÷ 2 = 2 with a remainder of 1. So the answer of 5 % 2 = 1. Example #2, 23 mod 5 means 23 ÷ 5, which is 4, and the remainder is 3. That means, 23 mod 5 = 3. To answer these questions in the video, you will need all three numbers a,b and c. If you are not provided with the number c, then you will need to ask the person who assigned the exercise question.

[@quetzaltpa4450]

@@aborne thank you!

[@YT.ASMR.DRW.]

thank you ❤

[@luxjoseph2596]

why I feel like having Ross in my head? LOL

[@lmatton1]

can you explain the parametric equation a bit further?

[@mora8181]

Thanks, extremely good

[@beatrizrosa6189]

Thank you so much!

[@haru_ne_]

Thanks I love you

[@mixinfo6202]

Thank you so much

[@chandler200]

greattttttttttt👌 Thanks a lottttttttt 💓

[@aborne]

Glad this video helped you!

[@senadkhalfa8374]

thanks

[@ivayloivanov5766]

Can you help me with this, please: x ≡ 2 (mod 11) x ≡ 9 (mod 15) x ≡ 7 (mod 9) x ≡ 5 (mod 7) ?

[@aborne]

Those are some good ones. I advise you approach your instructor, Teaching Assistant, or professor on help with those.

[@sujaynaik1320]

Chinese remainder theorem?

[@ivayloivanov5766]

@@sujaynaik1320 There is a problem in the second row.

[@ivayloivanov5766]

I took the exams. I hoop soon I will have a time and I'll write the solution.

[@sujaynaik1320]

@@ivayloivanov5766 okay!!

[@user-fr5be6gy3u]

Thank you!

[@AyushPatel]

THANKS!!!!!

[@tazkiakamalia]

thanks for this

[@nigelkay6264]

if you write an audio book on math you will definitely help infinity pips, and you will get some serious money

[@aborne]

Thank you for those kind words. What’s are infinity pips?

[@nigelkay6264]

@@aborne ok, i meant you could help infinity people 👌👍

[@The.mohisa]

Amazing 🤌🏼✨

[@miraculix666]

"One of the numbers is prime? The GCD is 1." Not true. GCD(prime, n*prime)=prime1

[@aborne]

That’s a good point.

[@emmaruhcheva3658]

lifesaver..

[@robertoberidojr.435]

Real good

[@kairiannah]

THE ABSOULE BEST

[@nasserkhamis1690]

Thank you so much may I get ur email

[@aborne]

Go to www.andyborne.com/math and you will find it there.

[@ciromaiello4111]

I love you

[@nasserkhamis1690]

Dear : could you check your e-mail

[@vishaldonda]

BAYES theorem.

[@tsunningwah3471]

no

[@Marryatau]

I am a primate

[@Ahnos]

I STILL DONT IT IM SO DUMB

[@aborne]

Don’t worry. Lots of people don’t getting this and they end up fine in life.

[@Ahnos]

@@aborne I have exam today that's why I'm trying to push this thing to my brain but it just doesn't go in :(

[@piotrek7633]

I dont understand any of this, books are too complicated if your basics arent correct and there isnt any material in simple terms. Kinda doomed im hoping to memorise everything and get it over with..

[@aborne]

Don’t give up on yourself. Start small and slowly work in more difficult examples.

[@MuhammadJamil-ed9cp]

Nice from pakistant

[@monjasonsteng7861]

Thank you so much

[@stamfordpolytechnic2656]

Thank you!