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

26 Mar, 2021
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++
Java
Python3
C#
PHP
Javascript
// C++ program to count natural numbers whose// factorials are divisible by x but not y.#include<bits/stdc++.h>using namespace std; // GCD function to compute the greatest// divisor among a and bint 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 divisible 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 codeint main(void){    int x = 15, y = 25;    cout << countFactorialXNotY(x, y);    return 0;}
// Java program to count natural numbers whose// factorials are divisible 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 divisible 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.
# Python program to count natural# numbers whose factorials are# divisible by x but not y. # GCD function to compute the greatest# divisor among a and bdef 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 divisible 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 codex = 15y = 25 print(countFactorialXNotY(x, y)) # This code is contributed by Anant Agarwal.
// C# program to count natural numbers whose// factorials are divisible 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 divisible 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.
<?php// PHP program to count natural// numbers whose factorials are// divisible by x but not y. // GCD function to compute the// greatest divisor among a and bfunction 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 $i;} // Count of natural numbers// whose factorials are divisible// by x but not y.function countFactorialXNotY($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.?>
<script> // Javascript program to Merge two sorted halves of// array Into Single Sorted Array     // 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)    {        let i = 1; // Result        let 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 divisible 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            let x = 15, y = 25;        document.write(countFactorialXNotY(x, y));                     </script>
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.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
nitin mittal
jit_t
ManasChhabra2
splevel62
divisibility
factorial
GCD-LCM
Mathematical
Mathematical
factorial
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Comments
Old Comments
Program to find sum of elements in a given array
Modulo Operator (%) in C/C++ with Examples
The Knight's tour problem | Backtracking-1
Algorithm to solve Rubik's Cube
Merge two sorted arrays
Euclidean algorithms (Basic and Extended)
Modular multiplicative inverse
Sieve of Eratosthenes
Program to find GCD or HCF of two numbers
Find minimum number of coins that make a given value