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

• Difficulty Level : Medium
• Last Updated : 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++

 `// C++ program to count natural numbers whose``// factorials are divisible by x but not y.``#include``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

## Java

 `// 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.`

## Python3

 `# 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 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 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 code``x ``=` `15``y ``=` `25` `print``(countFactorialXNotY(x, y))` `# This code is contributed by Anant Agarwal.`

## C#

 `// 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

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## Javascript

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Output :

`5`

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