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Minimum number of Factorials whose sum is equal to N
• Last Updated : 08 Apr, 2021

Given a number N (<1010), the task is to find the minimum number of factorials needed to represent N, as their sum. Also, print those factorials.
Examples:

```Input: N = 30
Output: 2
24, 6
Explanation:
Factorials needed to represent 30: 24, 6

Input: N = 150
Output: 3
120, 24, 6
Explanation:
Factorials needed to represent 150: 120 24 6```

Approach

1. In order to efficiently find the factorials which are needed to represent N as their sum, we can precompute the factorials till N (N < 1010) and store them in an array, for faster calculations.

2. Then using Greedy Algorithm, we can take largest factorials from this array which can be added to represent N.

3. Start from largest possible factorial and keep adding factorials while the remaining value is greater than 0.

4. Below is the complete algorithm.
• Initialize result as empty
• find the largest factorial that is smaller than N
• Add found factorial to result. Subtract value of found factorial from N
• If N becomes 0, then print result. Else repeat steps 2 and 3 for new value of N

Below is the implementation of the above approach:

## C++

 `// C++ program to find minimum number of factorials` `#include ``#define ll long long int``using` `namespace` `std;` `// Array to calculate all factorials``// less than or equal to N``// Since there can be only 14 factorials``// till 10^10``// Hence the maximum size of fact[] is 14``ll fact;` `// Store the actual size of fact[]``int` `size = 1;` `// Function to precompute factorials till N``void` `preCompute(``int` `N)``{``    ``// Precomputing factorials``    ``fact = 1;` `    ``for` `(``int` `i = 1; fact[i - 1] <= N; i++) {``        ``fact[i] = (fact[i - 1] * i);``        ``size++;``    ``}``}` `// Function to find the minimum number``// of factorials whose sum represents N``void` `findMin(``int` `N)``{` `    ``// Precompute factorials``    ``preCompute(N);` `    ``int` `originalN = N;` `    ``// Initialize result``    ``vector<``int``> ans;` `    ``// Traverse through all factorials``    ``for` `(``int` `i = size - 1; i >= 0; i--) {` `        ``// Find factorials``        ``while` `(N >= fact[i]) {``            ``N -= fact[i];``            ``ans.push_back(fact[i]);``        ``}``    ``}` `    ``// Print min count``    ``cout << ans.size() << ``"\n"``;` `    ``// Print result``    ``for` `(``int` `i = 0; i < ans.size(); i++)``        ``cout << ans[i] << ``" "``;``}` `// Driver program``int` `main()``{``    ``int` `n = 27;``    ``findMin(n);``    ``return` `0;``}`

## Java

 `// Java program to find minimum number of factorials``import` `java.util.*;` `class` `GFG{`` ` `// Array to calculate all factorials``// less than or equal to N``// Since there can be only 14 factorials``// till 10^10``// Hence the maximum size of fact[] is 14``static` `int` `[]fact = ``new` `int``[``14``];`` ` `// Store the actual size of fact[]``static` `int` `size = ``1``;`` ` `// Function to precompute factorials till N``static` `void` `preCompute(``int` `N)``{``    ``// Precomputing factorials``    ``fact[``0``] = ``1``;`` ` `    ``for` `(``int` `i = ``1``; fact[i - ``1``] <= N; i++) {``        ``fact[i] = (fact[i - ``1``] * i);``        ``size++;``    ``}``}`` ` `// Function to find the minimum number``// of factorials whose sum represents N``static` `void` `findMin(``int` `N)``{`` ` `    ``// Precompute factorials``    ``preCompute(N);`` ` `    ``int` `originalN = N;`` ` `    ``// Initialize result``    ``Vector ans = ``new` `Vector();`` ` `    ``// Traverse through all factorials``    ``for` `(``int` `i = size - ``1``; i >= ``0``; i--) {`` ` `        ``// Find factorials``        ``while` `(N >= fact[i]) {``            ``N -= fact[i];``            ``ans.add(fact[i]);``        ``}``    ``}`` ` `    ``// Print min count``    ``System.out.print(ans.size()+ ``"\n"``);`` ` `    ``// Print result``    ``for` `(``int` `i = ``0``; i < ans.size(); i++)``        ``System.out.print(ans.get(i)+ ``" "``);``}`` ` `// Driver program``public` `static` `void` `main(String[] args)``{``    ``int` `n = ``27``;``    ``findMin(n);``}``}` `// This code is contributed by PrinciRaj1992`

## Python3

 `# Python3 program to find minimum number of factorials` `# Array to calculate all factorials``# less than or equal to N``# Since there can be only 14 factorials``# till 10^10``# Hence the maximum size of fact[] is 14``fact ``=` `[``0``]``*``14` `# Store the actual size of fact[]``size ``=` `1` `# Function to precompute factorials till N``def` `preCompute(N):``    ``global` `size``    ` `    ``# Precomputing factorials``    ``fact[``0``] ``=` `1` `    ``i ``=` `1` `    ``while` `fact[i ``-` `1``] <``=` `N:``        ``fact[i] ``=` `fact[i ``-` `1``] ``*` `i``        ``size ``+``=` `1``        ``i ``+``=` `1` `# Function to find the minimum number``# of factorials whose sum represents N``def` `findMin(N):` `    ``# Precompute factorials``    ` `    ``preCompute(N)` `    ``originalN ``=` `N` `    ``# Initialize result``    ``ans ``=` `[]` `    ``# Traverse through all factorials``    ``for` `i ``in` `range``(size``-``1``, ``-``1``, ``-``1``):` `        ``# Find factorials``        ``while` `(N >``=` `fact[i]):``            ``N ``-``=` `fact[i]``            ``ans.append(fact[i])` `    ``# Prmin count``    ``print``(``len``(ans))` `    ``# Prresult``    ``for` `i ``in` `ans:``        ``print``(i, end``=``" "``)` `# Driver program` `n ``=` `27``findMin(n)` `# This code is contributed by mohit kumar 29`

## C#

 `// C# program to find minimum number of factorials``using` `System;``using` `System.Collections.Generic;` `class` `GFG{``  ` `// Array to calculate all factorials``// less than or equal to N``// Since there can be only 14 factorials``// till 10^10``// Hence the maximum size of fact[] is 14``static` `int` `[]fact = ``new` `int``;``  ` `// Store the actual size of fact[]``static` `int` `size = 1;``  ` `// Function to precompute factorials till N``static` `void` `preCompute(``int` `N)``{``    ``// Precomputing factorials``    ``fact = 1;``  ` `    ``for` `(``int` `i = 1; fact[i - 1] <= N; i++) {``        ``fact[i] = (fact[i - 1] * i);``        ``size++;``    ``}``}``  ` `// Function to find the minimum number``// of factorials whose sum represents N``static` `void` `findMin(``int` `N)``{``  ` `    ``// Precompute factorials``    ``preCompute(N);``  ` `    ``int` `originalN = N;``  ` `    ``// Initialize result``    ``List<``int``> ans = ``new` `List<``int``>();``  ` `    ``// Traverse through all factorials``    ``for` `(``int` `i = size - 1; i >= 0; i--) {``  ` `        ``// Find factorials``        ``while` `(N >= fact[i]) {``            ``N -= fact[i];``            ``ans.Add(fact[i]);``        ``}``    ``}``  ` `    ``// Print min count``    ``Console.Write(ans.Count+ ``"\n"``);``  ` `    ``// Print result``    ``for` `(``int` `i = 0; i < ans.Count; i++)``        ``Console.Write(ans[i]+ ``" "``);``}``  ` `// Driver program``public` `static` `void` `Main(String[] args)``{``    ``int` `n = 27;``    ``findMin(n);``}``}` `// This code is contributed by PrinciRaj1992`

## Javascript

 ``
Output:
```3
24 2 1```

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