Number Theorem | Gauss' Theorem

We prove Gauss's Theorem. That is, we prove that the sum of values of the Euler phi function over divisors of n is equal to n.

[georgesadler]

Professor Penn, thank you for a fantastic introduction to Gauss' Theorem.

[hybmnzz]

At 4:10 it should mean the cardinality of Sd is equal to the function at (n/d). You correct yourself later.

[balthazarbeutelwolf]

there was an issue on the first board with m; there was no guarantee that m is a multiple of d, so things such as gcd(m/d, whatever) are not well defined.

[jasonsusayajps]

Can we have a PDF link for this sir ? Pleas. Badly needed for my practice teach. 🥺 I need your help sir..

[alexandrebailly]

There is an issue.
Sd = { 1<= m <= n | (d | m) and gcd(m/d, n/d) }
Sk = { 1<= k <= n/d | gcd(k, n/d) }
Sd and Sk are not the same.
E.g., n = 12, d = 2
Sd = { 2, 10 }
Sk = { 1, 5 }
But #Sd = #Sk

[peelysl]

Good for preparing for mathematical olympiads

[TechToppers]

Old Michael sir is really an example of super rigorous mathematician giving just plain the logic and saying "its clear that" just like a very normal mathematician.