The document discusses the Euler Phi function, which calculates the number of numbers less than n that are relatively prime to n. It provides examples of calculating phi(n) for different types of numbers n. For prime numbers, phi(n) = n-1. For numbers that are a power of a prime like 8, phi(n) = n - n/p, where n is the power of p. For numbers that can be expressed as a product of different primes, phi(n) is the product of phi(n) for each prime factor.

1. Euler Phi
Lets talk about the problem first,
we want to know how many number is relatively prime / co-primes to N.
What's relatively prime / Co-prime ?
if gcd(x,y) == 1 , then they are x and y are co-primes.
Simply,
if 1 is the only number that can evenly divide x and y then they are
co-primes.
As example,
So, 12 and 13 are co-primes.
Our problem was,
how many numbers are there which are co-primes to N.
Means,
Lets Assume , N = 8
We can see there are 4 number co-primes with N=8.
So we are looking for a general formula to find for any N.

2. Euler phi function do that for us.
Suppose we are going to find the value of phi(N)
where,
Here p1 , p2 .. pk are primes
and this is actually the prime factorization of N
So for N = 24
Lets talk about the formula to find out the phi() value for any N.
formula is,
Lets try the proof of it,
Lets talk about some case,
Case 1 : N is Prime
When N is primes we can make it sure that there is no number
except 1 and itself that can divide it otherwise it cant be a prime.
So , it maintained such relation with every number less than it.
the relation is,
gcd(x,n) = 1;
So we can surely say that phi of such n is phi(n) = n-1.
When n is a prime of course.

3. Case 2 : When N is power of Prime
Means if we can express N as power of 1 single prime.
Like N = 8
we can write is as N = 2 x 2 x 2.
or N = 2 ^ 3
In this scenario,
phi(n) = n - (n / p);
where n = p ^ x; =======================> x is any integer,p is prime <===
HOW !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Lets take an example,
where N = 8.
So , every number which are divisble by 2 before 8 are not maintain
the realtion.
which is,
gcd(n,x) 1;
bc their gcd will be 2 or greater than 2.

4. Here we can see , 4 numbers are not coprimes to n.
which is exactly
n / p = 8 / 2 = 4.
so if we subtract all those numbers whose are not co-prime to N
then we could be able to find the exact value of phi(n).
in this case that is,
phi(8) = 8 - (8 / 2) = 4
so phi(8) = 4;
if we can write N like this ,
Then the number of numbers not coprime to N is ,
Which are actually divisible by P.So they cant be coprime to N.
so the final foumula for such N that can be written as a single
prime is,
Now its looks like the equation above.The general formula,

5. We know Phi function is multiplicative,
means,
phi(m * n) = phi(m) * phi(n).
Lets not talk about the proof of multiplicity of phi now.
Assume that is multiplicative and complete the proof.
So as we know any number can express multiplication of primes.What we
call primes factorization.
lets say,factorization of N is,
We already found out that phi of such numbers that can be expressed
as power of single prime is,

6. Now its look like exactly the same we written before.
You can follow the reference list below to know further.
Thank you.