Introduction to Euler's Totient Function!

[@xxxprawn8374]

I was researching this when I was preparing for olympiads this year :) Very well explained!! Keep it up!

[@castrojosua]

This actually just changed my life

[@chrisgreen_1729]

I love your videos! They're so clear and well-explained.

[@hhzz879]

How clear it is! Thanks a lot.

[@datsmydab-minecraft-and-mo5666]

Wonderfully explained

[@xinxialiu2651]

Very clear and helpful! Thank you!

[@xoppa09]

nice proof! very intuitive.

[@aimeerimturatbekova242]

It was very helpful, thanks a lot!!!!

[@mohak4275]

Very interesting function

[@klaybond7739]

I wished my math teacher had explained it like this

[@mohammedno5222]

Thank you so much for this video it help me :)

[@JohnSmith-xv1xy]

Great video! Very good explanation. Can you please clarify one example: Let's say we have 3^6=729. According to the video, we should have 3^5=243 integers (less than or equal to 3^6) that have common divisor/factor with 729 (basically, multiples of 3). Does it mean that the number of co-prime integers is 729-243=486?

[@MuPrimeMath]

Yes, the number of integers in the interval [1, 729] that are coprime to 729 is 486 by the reasoning you gave.

[@JohnSmith-xv1xy]

@@MuPrimeMath Thanks a lot!

[@dvoorganisation459]

THANK YOU SO MUCH SIR 🙏🙏🙏

[@aashsyed1277]

great proof!

[@AaronHollander314]

Why do we consider n as having a property that it can never have? The first criteria should be to check all numbers up to n-1.

[@MuPrimeMath]

The totient function of 1 is 1 because 1 is coprime to itself. That's why.

[@jisancy5423]

THANK YOU SO MUCH

[@premkashyap9579]

Thank you

[@ZackSussmanMusic]

Why do we count 1 as a non-shared factor even when 5= 5*1 and 6 = 6*1

[@MuPrimeMath]

We don't consider 1 to be a shared factor because literally every whole number has 1 as a factor. At that point, talking about shared factors wouldn't be very useful!

[@ZackSussmanMusic]

Mu Prime Math thanks!

[@CrashExhibition]

I am missing one basic but crucial fact due to dumbness. We use P^(a-1) as the common factor, why is it not P^a? I understand how to factor it out and get the result, but I don’t really understand why we choose it as a common factor.

[@CrashExhibition]

I think it’s because it’s the lower exponent so we know we can factor it out of the higher exponent. It’s really weird how it gets confusing to me when we replace a numbered exponent with a variable 😅

[@MuhammadTaimur]

Shouldn't the prime factors of p^a be p^1, p^2, p^3......p^a instead of p, 2p, 3p, 4p...?

[@MuPrimeMath]

We aren't looking for divisors of p^a, but rather any numbers which share a common factor with p^a, which makes them not coprime to p^a.

[@jasonhuang5175]

It's a little hard to understand but still ok. Thank you!!

[@razvanbirisan9521]

Without Euler's product formula? 😥

[@MuPrimeMath]

I will cover some more advanced stuff in the next few videos!

[@reardelt]

@@MuPrimeMath Hey man do you have this video/

[@MuPrimeMath]

@@reardelt kzbin.info/www/bejne/fpi4l3WubbWpsJI

[@ebtisamhussien7389]

Thank you