Number Theory | Inverses modulo n

We give a characterization of numbers which are invertible modulo n.

[thinguyen]

Thanks you teacher Penn for dedicating time and energy to create awesome math videos that helps simplify math like this. It helps students around the world who struggle with math a lot.

[azharlatif]

Michael Penn is Super human embodying as Mathematician, Mountaineer, Diver and what not. Saw his Q n A footage to know him. Stay Blessed Michael !!!

[gorgzilla]

Thank you so much for being so clear and concise in all these videos; really helps a lot! Subscribed.

[georgesadler]

Professor Penn, thank you for fantastic analysis on Inverses modulo n. These are classic topics in Number Theory.

[ramakrishnasen]

Try taking off your earphones and watching this video! The magic is, you'll still understand what he wants to explain!!

[azizketata]

the tyoe of teacher who will fuckk you up if you're not attentive . Keep lifting man.

[MyOneFiftiethOfADollar]

Noticed for arbitrary modulus n that
(n-1)(n-1)==(-1)(-1)=1 (mod n)

Put another way, n-1 is it’s own inverse (mod n). Also consistent with n and n-1 being relatively prime.

Special case of this is what you showed
(8)(8)==1 (mod 9)

[Walczyk]

michael penn is doing the world a great service, really for anyone interested in pure math

[PunmasterSTP]

I know that you just discussed inverses, but I have to say that this video was singular. Thanks again for making all of these!

[kakalibiswas]

Thanks sir for giving free knowledge of number theory

[Ensivion]

so for only prime n, do each of the non-zero minimal residues have an actual 'modulo inverse', and n-1 always is its own inverse, I have a feeling this is very important when studying primes.

[hypnovia]

If you can have the inverse of a number mod n… what’s stopping there from being an inverse of a MATRIX mod n? What if det(M) mod n is equal to 0, or not relatively prime to n?

[tiandaochouqin]

Nice work. At 7:32, about the reverse direction proof, you said you proved it in the channel before, could you provide the link? Thanks!

[ren]

How do we prove that these inverse pairs are unique?

[chrisjuravich]

So influenced am I by these videos that I find myself saying “good“ after each time I make a true statement of fact.

[sgssergio]

Hi teacher, in 1:56 you write 2^-1 = 5 mod 9, 5^-1 cong 2 mod 9. In the first why do you use equal sumbol and not congruence symbol instead?

[mohammedsamir]

6:32
Could anyone explain this step for me?!

[kobethebeefinmathworld]

I'll say it's more like a ring theory problem

[jonatanorange]

Great content, but Bruh, in the future recordings please stop missing steps and call them "tricks".