What is Euler's Totient Function?
Number theory is one of the most important topics in the field of Math and can be used to solve a variety of problems. Many times one might have come across problems that relate to the prime factorization of a number, to the divisors of a number, to the multiples of a number and so on.
Euler's Totient function is a function that is related to getting the number of numbers that are coprime to a certain number X that are less than or equal to it. In short , for a certain number X we need to find the count of all numbers Y where gcd(X,Y)=1 and 1≤Y≤X.
A naive method to do so would be to Brute-Force the answer by checking the gcd of X and every number less than or equal to X and then incrementing the count whenever a GCD of 1 is obtained. However, this can be done in a much faster way using Euler's Totient Function.
According to Euler's product formula, the value of the Totient function is below the product over all prime factors of a number. This formula simply states that the value of the Totient function is the product after multiplying the number N by the product of (1−(1/p)) for each prime factor of N.
So,
ϕ(n)=n∏p prime p|n(1−1p)
Algorithm steps:
- Generate a list of primes.
- While dealing with a certain N, check and store all the primes that perfectly divide N.
- Now, it is just needed to use these primes and the above formula to get the result.
Implementation:
set<> primes;
static void mark(int num,int max,int[] arr)
{
int i=2,elem;
while((elem=(num*i))<=max)
{
arr[elem-1]=1;
i++;
}
}
GeneratePrimes()
{
int arr[max_prime];
for(int i=1;i<arr.length;i++)
{
if(arr[i]==0)
{
list.add(i+1);
mark(i+1,arr.length-1,arr);
}
}
}
main()
{
GeneratePrimes();
int N=nextInt();
int ans=N;
for(int k:set)
{
if(N%k==0)
{
ans*=(1-1/k);
}
}
print(ans);
}
There are a few subtle observations that one can make about Euler's Totient Function.
- The sum of all values of Totient Function of all divisors of N is equal to N.
- The value of Totient function for a certain prime P will always be P−1 as the number P will always have a GCD of 1 with all numbers less than or equal to it except itself.
- For 2 number A and B, if GCD(A,B)==1 then Totient(A)×Totient(B) = Totient(A⋅B).