Number Theory | The Multiplicativity of Euler's Totient Function

We state and prove when Euler's totient function is multiplicative.

[PunmasterSTP]

I'd say this is another phi-nomenal video; thanks again for making this wonderful series!

[PrinceKumar-yolr]

Sir very good vedio
From five days i was stuck with the proof but really
U cleared my doubt
Thanks a lot sir
💚💚Million love from india🌼🌻❤❤❤

[souverainer]

Hello Prof, I think your definition of totient is not quite right - the range of m should be up to n, i.e, m such that 1 <= m <= n and gcd(m, n)=1. Otherwise you don't get phi(1)=1

[georgesadler]

Professor M. Penn, thank you for an excellent lecture on The Multiplicativity of Euler's Totient Function.

[liyi-hua]

excellent explanation. that is exactly the argument i want to hear

[fromscratch]

I'm not sure I agree the statement that lm+k = l'm+k (mod n). Looking at two numbers from the same column in the example like 1 and 4, they are clearly not congruent modulo 5. Can you help understand this?

[sangamesh]

the phi of n part was tricky for me to understand....had to watch it i guess understood after using paper and thank if possible make videos using a white board that would be more clear to view.

[余淼-eb]

You are a true genius. Thanks so much for your sharing.

[Shrikant_Anand]

I don't understand why Penn mentioned that gcd of every number in any column with m is same at 6:35?

[paulwang]

Great lecture! But seems you it'll be clearer to use the the concepts of Complete Residue System (CRS) and Reduced CRS.

[TechToppers]

I've same proof in my textbook. But it was generalised immediately...
I'm preparing for Olympiads...

[ZEPHYRZHANG-mgzi]

lol he always sounds like he's out of breath and the botw background.

[tomatrix]

The finishing explanation I found unclear.