# 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
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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 divisble 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 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. `

## Python3

 `# 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. `

## C#

 `// 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. `

## PHP

 ` `

Output :

```5
```