Count natural numbers whose factorials are divisible by x but not y

Given two numbers x and y (x <= y), find out the total number of natural numbers, say i, for which i! is divisible by x but not y.

Examples :

Input : x = 2, y = 5
Output : 3
There are three numbers, 2, 3 and 4
whose factorials are divisible by x
but not y.

Input: x = 15, y = 25
Output: 5
5! = 120 % 15 = 0 && 120 % 25 != 0
6! = 720 % 15 = 0 && 720 % 25 != 0
7! = 5040 % 15 = 0 && 5040 % 25 != 0
8! = 40320 % 15 = 0 && 40320 % 25 != 0
9! = 362880 % 15 = 0 && 362880 % 25 != 0
So total count = 5

Input: x = 10, y = 15
Output: 0

For all numbers greater than or equal to y, their factorials are divisible by y. So all natural numbers to be counted must be less than y.

A simple solution is to iterate from 1 to y-1 and for every number i check if i! is divisible by x and not divisible by y. If we apply this naive approach, we wouldn’t go above 20! or 21! (long long int will have its upper limit)

A better solution is based on below post.
Find the first natural number whose factorial is divisible by x
We find the first natural numbers whose factorials are divisible by x! and y! using above approach. Let the first natural numbers whose factorials are divisible by x and y be xf and yf respectively. Our final answer would be yf – xf. This formula is based on the fact that if i! is divisible by a number x, then (i+1)!, (i+2)!, … are also divisible by x.

Below is the implementation.

C++

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// C++ program to count natural numbers whose
// factorials are divisble by x but not y.
#include<bits/stdc++.h>
using namespace std;
  
// GCD function to compute the greatest
// divisor among a and b
int gcd(int a, int b)
{
    if ((a % b) == 0)
        return b;
    return gcd(b, a % b);
}
  
// Returns first number whose factorial
// is divisible by x.
int firstFactorialDivisibleNumber(int x)
{
   int i = 1;  // Result
   int new_x = x;
  
   for (i=1; i<x; i++)
   {
       // Remove common factors
       new_x /= gcd(i, new_x);
  
       // We found first i.
       if (new_x == 1)
          break;
   }
   return i;
}
  
// Count of natural numbers whose factorials
// are divisble by x but not y.
int countFactorialXNotY(int x, int y)
{
    // Return difference between first natural
    // number whose factorial is divisible by
    // y and first natural number whose factorial
    // is divisible by x.
    return (firstFactorialDivisibleNumber(y) -
            firstFactorialDivisibleNumber(x));
}
  
// Driver code
int main(void)
{
    int x = 15, y = 25;
    cout << countFactorialXNotY(x, y);
    return 0;
}

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Java

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// Java program to count natural numbers whose
// factorials are divisble by x but not y.
  
class GFG
{
    // GCD function to compute the greatest
    // divisor among a and b
    static int gcd(int a, int b)
    {
        if ((a % b) == 0)
            return b;
        return gcd(b, a % b);
    }
      
    // Returns first number whose factorial
    // is divisible by x.
    static int firstFactorialDivisibleNumber(int x)
    {
        int i = 1; // Result
        int new_x = x;
          
        for (i = 1; i < x; i++)
        {
            // Remove common factors
            new_x /= gcd(i, new_x);
          
            // We found first i.
            if (new_x == 1)
                break;
        }
        return i;
    }
      
    // Count of natural numbers whose factorials
    // are divisble by x but not y.
    static int countFactorialXNotY(int x, int y)
    {
        // Return difference between first natural
        // number whose factorial is divisible by
        // y and first natural number whose factorial
        // is divisible by x.
        return (firstFactorialDivisibleNumber(y) -
                firstFactorialDivisibleNumber(x));
    
      
    // Driver code
    public static void main (String[] args)
    {
        int x = 15, y = 25;
        System.out.print(countFactorialXNotY(x, y));
    }
}
  
// This code is contributed by Anant Agarwal.

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Python3

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# Python program to count natural
# numbers whose factorials are 
# divisble by x but not y.
  
# GCD function to compute the greatest
# divisor among a and b
def gcd(a, b):
      
    if ((a % b) == 0):
        return b
      
    return gcd(b, a % b)
  
# Returns first number whose factorial
# is divisible by x.
def firstFactorialDivisibleNumber(x):
      
    # Result
    i = 1 
    new_x = x
      
    for i in range(1, x):
          
        # Remove common factors
        new_x /= gcd(i, new_x)
  
        # We found first i.
        if (new_x == 1):
            break
          
    return i
  
# Count of natural numbers whose 
# factorials are divisble by x but
# not y.
def countFactorialXNotY(x, y):
  
    # Return difference between first 
    # natural number whose factorial 
    # is divisible by y and first 
    # natural number whose factorial
    # is divisible by x.
    return (firstFactorialDivisibleNumber(y) -
            firstFactorialDivisibleNumber(x))
              
# Driver code
x = 15
y = 25
  
print(countFactorialXNotY(x, y))
  
# This code is contributed by Anant Agarwal.

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C#

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// C# program to count natural numbers whose
// factorials are divisble by x but not y.
using System;
  
class GFG
{
      
    // GCD function to compute the greatest
    // divisor among a and b
    static int gcd(int a, int b)
    {
        if ((a % b) == 0)
            return b;
        return gcd(b, a % b);
    }
      
    // Returns first number whose factorial
    // is divisible by x.
    static int firstFactorialDivisibleNumber(int x)
    {
        int i = 1; // Result
        int new_x = x;
          
        for (i = 1; i < x; i++)
        {
              
            // Remove common factors
            new_x /= gcd(i, new_x);
          
            // We found first i.
            if (new_x == 1)
                break;
        }
          
        return i;
    }
      
    // Count of natural numbers whose factorials
    // are divisble by x but not y.
    static int countFactorialXNotY(int x, int y)
    {
          
        // Return difference between first natural
        // number whose factorial is divisible by
        // y and first natural number whose factorial
        // is divisible by x.
        return (firstFactorialDivisibleNumber(y) -
                firstFactorialDivisibleNumber(x));
    
      
    // Driver code
    public static void Main ()
    {
        int x = 15, y = 25;
          
        Console.Write(countFactorialXNotY(x, y));
    }
}
  
// This code is contributed by nitin mittal.

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PHP

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<?php
// PHP program to count natural
// numbers whose factorials are
// divisble by x but not y.
  
// GCD function to compute the 
// greatest divisor among a and b
function gcd($a, $b)
{
    if (($a % $b) == 0)
        return $b;
    return gcd($b, $a % $b);
}
  
// Returns first number whose 
// factorial is divisible by x.
function firstFactorialDivisibleNumber($x)
{
    // Result
    $i = 1; 
    $new_x = $x;
      
    for ($i = 1; $i < $x; $i++)
    {
        // Remove common factors
        $new_x /= gcd($i, $new_x);
      
        // We found first i.
        if ($new_x == 1)
            break;
    }
    return $i;
}
  
// Count of natural numbers 
// whose factorials are divisble
// by x but not y.
function countFactorialXNotY($x, $y)
{
    // Return difference between
    // first natural number whose 
    // factorial is divisible by 
    // y and first natural number 
    // whose factorial is divisible by x.
    return (firstFactorialDivisibleNumber($y) -
            firstFactorialDivisibleNumber($x));
}
  
// Driver code
$x = 15; $y = 25;
echo(countFactorialXNotY($x, $y));
  
// This code is contributed by Ajit.
?>

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Output :

5

This article is contributed by Shubham Gupta. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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Improved By : nitin mittal, jit_t