Given an integer n, find the power of a given prime number(r) in n!
Input : n = 6 r = 3 Factorial of 6 is 720 = 2^4 * 3^2 *5 (prime factor 2,3,5) and power of 3 is 2 Output : 2 Input : n = 2000 r = 7 Output : 330
A simple method is to first calculate factorial of n, then factorize it to find the power of a prime number.
The above method can cause overflow for a slightly bigger numbers as factorial of a number is a big number. The idea is to consider prime factors of a factorial n.
Legendre Factorization of n!
For any prime number p and any integer n, let Vp(n) be the exponent of the largest power of p that divides n (that is, the p-adic valuation of n). Then
Vp(n) = summation of floor(n / p^i) and i goes from 1 to infinity
While the formula on the right side is an infinite sum, for any particular values of n and p it has only finitely many nonzero terms: for every i large enough that p^i > n, one has floor(n/p^i) = 0 . since, the sum is convergent.
Power of ‘r’ in n! = floor(n/r) + floor(n/r^2) + floor(n/r^3) + ....
Program to count power of a no. r in n! based on above formula.
- Finding power of prime number p in n!
- Find power of power under mod of a prime
- Largest power of k in n! (factorial) where k may not be prime
- Check if given number is a power of d where d is a power of 2
- Count occurrences of a prime number in the prime factorization of every element from the given range
- Check if a prime number can be expressed as sum of two Prime Numbers
- Find coordinates of a prime number in a Prime Spiral
- Highest power of a number that divides other number
- Number of factors of very large number N modulo M where M is any prime number
- Sum of digits of a given number to a given power
- Find whether a given number is a power of 4 or not
- GCD of a number raised to some power and another number
- Check if a number is a power of another number
- Highest power of 2 less than or equal to given number
- Check if a number can be expressed as power | Set 2 (Using Log)
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.
Improved By : jit_t