Smallest number S such that N is a factor of S factorial or S!

Given a number N. You are tasked with finding the smallest number S, such that N is a factor of S! (S factorial). N can be very large.

Examples:

Input  : 6
Output : 3
The value of 3! is 6
This is the smallest number which can have 6 as a factor.

Input  : 997587429953
Output : 998957
If we calculate out 998957!, 
we shall find that it is divisible by 997587429953.
Factors of 997587429953 are 998957 and 998629.

Naive Approach
We iterate from 1 to N, calculating factorial in each case. When we find a factorial that’s capable of having N as a factor, we output it. This method will be difficult to implement for large N, as the factorial can become very large.
Time Complexity: O(N^2)



Optimized Naive Approach
Instead of iterating from 1 to N, we use binary search. This is still a bad method, as we are still trying to calculate N!
Time Complexity O(N log N)

Optimum Solution
We can first calculate all the prime factors of N. We then reduce our problem to finding a factorial which has all the prime factors of N, at least as many times as they appear in N. We then binary search on elements from 1 to N. We can utilize Legendre’s Formula to check whether a number’s factorial has all the same prime factors. We then find the smallest such number.

C++

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// Program to find factorial that N belongs to
#include <bits/stdc++.h>
using namespace std;
  
#define ull unsigned long long
  
// Calculate prime factors for a given number
map<ull, int> primeFactors(ull num)
{
    // Container for prime factors
    map<ull, int> ans;
  
    // Iterate from 2 to i^2 finding all factors
    for (ull i = 2; i * i <= num; i++)
    {
        while (num % i == 0)
        {
            num /= i;
            ans[i]++;
        }
    }
  
    // If we still have a remainder
    // it is also a prime factor
    if (num > 1)
        ans[num]++;
    return ans;
}
  
// Calculate occurence of an element
// in factorial of a number
ull legendre(ull factor, ull num)
{
    ull count = 0, fac2 = factor;
    while (num >= factor)
    {
        count += num / factor;
        factor *= fac2;
    }
    return count;
}
  
bool possible(map<ull, int> &factors, ull num)
{
    // Iterate through prime factors
    for (map<ull, int>::iterator it = factors.begin();
            it != factors.end(); ++it)
    {
        // Check if factorial contains less
        // occurences of prime factor
        if (legendre(it->first, num) < it->second)
            return false;
    }
    return true;
}
  
// Function to binary search 1 to N
ull search(ull start, ull end, map<ull, int> &factors)
{
    ull mid = (start + end) / 2;
  
    // Prime factors are not in the factorial
    // Increase the lowerbound
    if (!possible(factors, mid))
        return search(mid + 1, end, factors);
  
    // We have reached smallest occurrence
    if (start == mid)
        return mid;
  
    // Smaller factorial satisfying
    // requirements may exist, decrease upperbound
    return search(start, mid, factors);
}
  
// Calculate prime factors and search
ull findFact(ull num)
{
    map<ull, int> factors = primeFactors(num);
    return search(1, num, factors);
}
  
// Driver function
int main()
{
    cout << findFact(6) << "n";
    cout << findFact(997587429953) << "n";
    return 0;
}

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Java

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// Java Program to find factorial that N belongs to
  
import java.util.HashMap;
import java.util.Iterator;
import java.util.Set;
  
class Test
{
    // Calculate prime factors for a given number
    static HashMap<Long, Integer> primeFactors(long num)
    {
          
        // Container for prime factors
        HashMap<Long, Integer> ans = new HashMap<Long, Integer>(){
            @Override
            public Integer get(Object key) {
                if(containsKey(key)){
                    return super.get(key);                          
            }
            return 0;
        }
    };
  
       
        // Iterate from 2 to i^2 finding all factors
        for (long i = 2; i * i <= num; i++)
        {
            while (num % i == 0)
            {
                num /= i;
                ans.put(i, ans.get(i)+1);
            }
        }
       
        // If we still have a remainder
        // it is also a prime factor
        if (num > 1)
            ans.put(num, ans.get(num)+1);;
        return ans;
    }
      
    // Calculate occurence of an element
    // in factorial of a number
    static long legendre(long factor, long num)
    {
        long count = 0, fac2 = factor;
        while (num >= factor)
        {
            count += num / factor;
            factor *= fac2;
        }
        return count;
    }
       
    static boolean possible(HashMap<Long, Integer> factors, long num)
    {
        Set<Long> s = factors.keySet();
          
        // Iterate through prime factors
        Iterator<Long> itr = s.iterator();
          
        while (itr.hasNext()) {
            long temp = itr.next();
             // Check if factorial contains less
            // occurrences of prime factor
            if (legendre(temp, num) < factors.get(temp))
                return false;
        }
          
        return true;
    }
       
    // Method to binary search 1 to N
    static long search(long start, long end, HashMap<Long, Integer> factors)
    {
        long mid = (start + end) / 2;
       
        // Prime factors are not in the factorial
        // Increase the lowerbound
        if (!possible(factors, mid))
            return search(mid + 1, end, factors);
       
        // We have reached smallest occurrence
        if (start == mid)
            return mid;
       
        // Smaller factorial satisfying
        // requirements may exist, decrease upperbound
        return search(start, mid, factors);
    }
       
    // Calculate prime factors and search
    static long findFact(long num)
    {
        HashMap<Long, Integer>  factors = primeFactors(num);
        return search(1, num, factors);
    }
      
    // Driver method
    public static void main(String args[])
    {
        System.out.println(findFact(6));
        System.out.println(findFact(997587429953L));
    }
}
// This code is contributed by Gaurav Miglani

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Output:

3
998957

At no point do we actually calculate a factorial. This means we do not have to worry about the factorial being too large to store.
Lagrange’s Formula runs in O(Log N).
Binary search is O(Log N).
Calculating prime factors is O(sqrt(N))
Iterating through prime factors is O(Log N).

Time complexity becomes: O(sqrt(N) + (Log N)^3)

This article is contributed by Aditya Kamath. 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.

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