# 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

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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<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 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 code` `int` `main(` `void` `)` `{` ` ` `int` `x = 15, y = 25;` ` ` `cout << countFactorialXNotY(x, y);` ` ` `return` `0;` `}` |

## 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

`<?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 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` `$i` `;` `}` `// Count of natural numbers` `// whose factorials are divisible` `// by x but not y.` `function` `countFactorialXNotY(` `$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.` `?>` |

## Javascript

`<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

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