Find the first natural number whose factorial is divisible by x
Given a number x, the task is to find first natural number i whose factorial is divisible by x.
Examples :
Input : x = 10 Output : 5 5 is the smallest number such that (5!) % 10 = 0 Input : x = 16 Output : 6 6 is the smallest number such that (6!) % 16 = 0
A simple solution is to iterate from 1 to x-1 and for every number i check if i! is divisible by x.
C++
// A simple C++ program to find first natural // number whose factorial divides x. #include <bits/stdc++.h> using namespace std; // Returns first number whose factorial // divides x. int firstFactorialDivisibleNumber(int x) { int i = 1; // Result int fact = 1; for (i = 1; i < x; i++) { fact = fact * i; if (fact % x == 0) break; } return i; } // Driver code int main(void) { int x = 16; cout << firstFactorialDivisibleNumber(x); return 0; } |
Java
// A simple Java program to find first natural // number whose factorial divides x class GFG { // Returns first number whose factorial // divides x. static int firstFactorialDivisibleNumber(int x) { int i = 1; // Result int fact = 1; for (i = 1; i < x; i++) { fact = fact * i; if (fact % x == 0) break; } return i; } // Driver code public static void main(String[] args) { int x = 16; System.out.print(firstFactorialDivisibleNumber(x)); } } // This code is contributed by Anant Agarwal. |
Python3
# A simple python program to find # first natural number whose # factorial divides x. # Returns first number whose # factorial divides x. def firstFactorialDivisibleNumber(x): i = 1; # Result fact = 1; for i in range(1, x): fact = fact * i if (fact % x == 0): break return i # Driver code x = 16print(firstFactorialDivisibleNumber(x)) # This code is contributed # by 29AjayKumar |
C#
// A simple C# program to find first natural // number whose factorial divides x using System; class GFG { // Returns first number whose factorial // divides x. static int firstFactorialDivisibleNumber(int x) { int i = 1; // Result int fact = 1; for (i = 1; i < x; i++) { fact = fact * i; if (fact % x == 0) break; } return i; } // Driver code public static void Main() { int x = 16; Console.Write( firstFactorialDivisibleNumber(x)); } } // This code is contributed by nitin mittal |
PHP
<?php // A simple PHP program to find // first natural number whose // factorial divides x. // Returns first number whose // factorial divides x. function firstFactorialDivisibleNumber($x) { // Result $i = 1; $fact = 1; for ($i = 1; $i < $x; $i++) { $fact = $fact * $i; if ($fact % $x == 0) break; } return $i; } // Driver code $x = 16; echo(firstFactorialDivisibleNumber($x)); // This code is contributed by Ajit. ?> |
Output :
6
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 avoids overflow. The solution is based on below observations.
- If i! is divisible by x, then (i+1)!, (i+2)!, … are also divisible by x.
- For a number x, all factorials i! are divisible by x when i >= x.
- If a number x is prime, then no factorial below x can divide it as x cannot be formed with multiplication of smaller numbers.
Below is algorithm
1) Run a loop for i = 1 to n-1
a) Remove common factors
new_x /= gcd(i, new_x);
b) Check if we found first i.
if (new_x == 1)
break;
2) Return i
Below is the implementation of above idea :
CPP
// C++ program to find first natural number // whose factorial divides x. #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 // divides 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; } // Driver code int main(void) { int x = 16; cout << firstFactorialDivisibleNumber(x); return 0; } |
Java
// Efficient Java program to find first // natural number whose factorial divides x. 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 // divides 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; } // Driver code public static void main(String[] args) { int x = 16; System.out.print(firstFactorialDivisibleNumber(x)); } } // This code is contributed by Anant Agarwal. |
Python3
# Python3 program to find first natural number # whose factorial divides x. # 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 # divides x. def firstFactorialDivisibleNumber(x): i = 1 # Result 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 # Driver code def main(): x = 16 print(firstFactorialDivisibleNumber(x)) if __name__ == '__main__': main() # This code is contributed by 29AjayKumar |
C#
// Efficient C# program to find first // natural number whose factorial // divides x. 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 divides 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; } // Driver code public static void Main() { int x = 16; Console.Write( firstFactorialDivisibleNumber(x)); } } // This code is contributed by nitin mittal. |
PHP
<?php // PHP program to find first // natural number whose // factorial divides x. // 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 % $b); } // Returns first number // whose factorial divides x. function firstFactorialDivisibleNumber($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; } // Driver code $x = 16; echo(firstFactorialDivisibleNumber($x)); // This code is contributed by Ajit. ?> |
Output :
6
Another approach using boost library:
(Thanking ajay0007 for contributing this approach)
Here we use boost library to efficiently calculate the value of factorial.
Prerequisite :boost-multiprecision-library
C++
// A cpp program for finding // the Special Factorial Number #include <bits/stdc++.h> #include <boost/multiprecision/cpp_int.hpp> using boost::multiprecision::cpp_int; using namespace std; // function for calculating factoial cpp_int fact(int n) { cpp_int num = 1; for (int i = 1; i <= n; i++) num = num * i; return num; } // function for check Special_Factorial_Number int Special_Factorial_Number(int k) { for(int i = 1 ; i <= k ; i++ ) { // call fact function and the // Modulo with k and check // if condition is TRUE then return i if ( ( fact (i) % k ) == 0 ) { return i; } } } //driver function int main() { // taking input int k = 16; cout<<Special_Factorial_Number(k); } |
Java
// Java program for finding // the Special Factorial Number public class GFG { // function for calculating factoial static int fact(int n) { int num = 1; for (int i = 1; i <= n; i++) { num = num * i; } return num; } // function for check Special_Factorial_Number static int Special_Factorial_Number(int k) { for (int i = 1; i <= k; i++) { // call fact function and the // Modulo with k and check // if condition is TRUE then return i if (fact(i) % k == 0) { return i; } } return 0; } //driver function public static void main(String[] args) { // taking input int k = 16; System.out.println(Special_Factorial_Number(k)); } } /*This code is contributed by Rajput-Ji*/ |
Python3
# Python 3 program for finding # the Special Factorial Number # function for calculating factoial def fact( n): num = 1 for i in range(1, n + 1): num = num * i return num # function for check Special_Factorial_Number def Special_Factorial_Number(k): for i in range(1, k + 1): # call fact function and the # Modulo with k and check # if condition is TRUE then return i if (fact(i) % k == 0): return i return 0 # Driver Code if __name__ == '__main__': # taking input k = 16 print(Special_Factorial_Number(k)) # This code is contributed by Rajput-Ji |
C#
// C# program for finding // the Special Factorial Number using System; public class GFG{ // function for calculating factoial static int fact(int n) { int num = 1; for (int i = 1; i <= n; i++) { num = num * i; } return num; } // function for check Special_Factorial_Number static int Special_Factorial_Number(int k) { for (int i = 1; i <= k; i++) { // call fact function and the // Modulo with k and check // if condition is TRUE then return i if (fact(i) % k == 0) { return i; } } return 0; } //driver function public static void Main() { // taking input int k = 16; Console.WriteLine(Special_Factorial_Number(k)); } } // This code is contributed by 29AjayKumar |
PHP
<?php // PHP program for finding // the Special Factorial Number // function for calculating // factoial function fact($n) { $num = 1; for ($i = 1; $i <= $n; $i++) $num = $num * $i; return $num; } // function for check // Special_Factorial_Number function Special_Factorial_Number($k) { for($i = 1 ; $i <= $k ; $i++ ) { // call fact function and the // Modulo with k and check // if condition is TRUE // then return i if (( fact ($i) % $k ) == 0 ) { return $i; } } } // Driver Code $k = 16; echo Special_Factorial_Number($k); // This code is contributed by Ajit. ?> |
Output :
6
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