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. |
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. |
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
CPP
// 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); } |
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
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.
Recommended Posts:
- Count numbers formed by given two digit with sum having given digits
- Smallest number with at least n trailing zeroes in factorial
- Sieve of Eratosthenes in 0(n) time complexity
- Find sum of divisors of all the divisors of a natural number
- Print the kth common factor of two numbers
- Find minimum number to be divided to make a number a perfect square
- Count natural numbers whose factorials are divisible by x but not y
- Find two prime numbers with given sum
- Circular primes less than n
- Primorial of a number
- Wilson's Theorem
- Check if a number can be expressed as x^y (x raised to power y)
- Count trailing zeroes in factorial of a number
- Efficient program to print all prime factors of a given number
- Sieve of Eratosthenes