# Print all combinations of factors (Ways to factorize)

Write a program to print all the combinations of factors of given number n.

Examples:

```Input : 16
Output :2 2 2 2
2 2 4
2 8
4 4

Input : 12
Output : 2 2 3
2 6
3 4```

To solve this problem we take one array of array of integers or list of list of integers to store all the factors combination possible for the given n. So, to achieve this we can have one recursive function which can store the factors combination in each of its iteration. And each of those list should be stored in the final result list.

Below is the implementation of the above approach.

## C++

 `#include ` `#include `   `using` `namespace` `std;`   `// Function to find all factor combinations of a number` `void` `backtrack(``int` `start, ``int` `target, vector<``int``>& factors,` `               ``vector >& combinations, ``int` `n)` `{` `    ``// Base case: if target is 1, we have found a` `    ``// factorization` `    ``if` `(target == 1) {` `        ``// Add a copy of the factors vector to combinations,` `        ``// except if the vector contains only one factor` `        ``// equal to n` `        ``if` `(factors.size() > 1 || factors[0] != n) {` `            ``combinations.push_back(factors);` `        ``}` `        ``return``;` `    ``}`   `    ``// Try all factors from start to target, inclusive` `    ``for` `(``int` `i = start; i <= target; i++) {` `        ``if` `(target % i == 0) {` `            ``factors.push_back(i); ``// Add i to factors` `            ``backtrack(i, target / i, factors, combinations,` `                      ``n); ``// Recursively find factors of` `                          ``// target / i` `            ``factors.pop_back(); ``// Remove i from factors` `        ``}` `    ``}` `}`   `// Function to call backtrack and return the factor` `// combinations` `vector > factorCombinations(``int` `n)` `{` `    ``vector > combinations;` `    ``vector<``int``> factors;` `    ``backtrack(2, n, factors, combinations, n);` `    ``return` `combinations;` `}`   `int` `main()` `{` `    ``int` `n = 12;` `    ``vector > combinations` `        ``= factorCombinations(n);`   `    ``cout << ``"All the combinations of factors of "` `<< n` `         ``<< ``" are:"` `<< endl;` `    ``for` `(vector<``int``> combination : combinations) {` `        ``for` `(``int` `factor : combination) {` `            ``cout << factor << ``" "``;` `        ``}` `        ``cout << endl;` `    ``}`   `    ``return` `0;` `}`

## Java

 `import` `java.util.ArrayList;` `import` `java.util.List;`   `public` `class` `FactorCombinations {`   `    ``public` `static` `List >` `    ``factorCombinations(``int` `n)` `    ``{` `        ``List > combinations` `            ``= ``new` `ArrayList<>();`   `        ``backtrack(``2``, n, ``new` `ArrayList<>(), combinations, n);`   `        ``return` `combinations;` `    ``}`   `    ``private` `static` `void` `    ``backtrack(``int` `start, ``int` `target, List factors,` `              ``List > combinations, ``int` `n)` `    ``{` `        ``if` `(target == ``1``) {` `            ``// add a copy of the factors list to` `            ``// combinations, except if the list contains` `            ``// only one factor equal to n` `            ``if` `(factors.size() > ``1` `|| factors.get(``0``) != n) {` `                ``combinations.add(``new` `ArrayList<>(factors));` `            ``}` `            ``return``;` `        ``}`   `        ``// try all factors from start to target, inclusive` `        ``for` `(``int` `i = start; i <= target; i++) {` `            ``if` `(target % i == ``0``) {` `                ``factors.add(i);` `                ``backtrack(i, target / i, factors,` `                          ``combinations, n);` `                ``factors.remove(factors.size() - ``1``);` `            ``}` `        ``}` `    ``}`   `    ``public` `static` `void` `main(String[] args)` `    ``{` `        ``int` `n = ``12``;` `        ``List > combinations` `            ``= factorCombinations(n);`   `        ``System.out.printf(` `            ``"All the combinations of factors of %d are:%n"``,` `            ``n);` `        ``for` `(List combination : combinations) {` `            ``System.out.println(combination);` `        ``}` `    ``}` `}`

## Python

 `def` `backtrack(start, target, factors, combinations, n):` `    ``# Base case: if target is 1, we have found a factorization` `    ``if` `target ``=``=` `1``:` `        ``# Add a copy of the factors list to combinations, except if the list contains only one factor equal to n` `        ``if` `len``(factors) > ``1` `or` `factors[``0``] !``=` `n:` `            ``combinations.append(``list``(factors))` `        ``return`   `    ``# Try all factors from start to target, inclusive` `    ``for` `i ``in` `range``(start, target ``+` `1``):` `        ``if` `target ``%` `i ``=``=` `0``:` `            ``factors.append(i)  ``# Add i to factors` `            ``# Recursively find factors of target // i` `            ``backtrack(i, target ``/``/` `i, factors, combinations, n)` `            ``factors.pop()  ``# Remove i from factors`     `def` `factorCombinations(n):` `    ``combinations ``=` `[]` `    ``factors ``=` `[]` `    ``backtrack(``2``, n, factors, combinations, n)` `    ``return` `combinations`     `if` `__name__ ``=``=` `'__main__'``:` `    ``n ``=` `12` `    ``combinations ``=` `factorCombinations(n)` `    ``print``(``"All the combinations of factors of {} are:"``.``format``(n))` `    ``for` `combination ``in` `combinations:` `        ``print``(combination)`

## C#

 `using` `System;` `using` `System.Collections.Generic;`   `namespace` `FactorCombinations {` `class` `Program {` `    ``static` `void` `Main(``string``[] args)` `    ``{` `        ``int` `n = 12;` `        ``List > combinations` `            ``= FactorCombinations(n);`   `        ``Console.WriteLine(` `            ``\$` `            ``"All the combinations of factors of {n} are:"``);` `        ``foreach``(List<``int``> combination ``in` `combinations)` `        ``{` `            ``Console.WriteLine(` `                ``string``.Join(``" "``, combination));` `        ``}` `    ``}`   `    ``static` `List > FactorCombinations(``int` `n)` `    ``{` `        ``List > combinations` `            ``= ``new` `List >();`   `        ``Backtrack(2, n, ``new` `List<``int``>(), combinations, n);`   `        ``return` `combinations;` `    ``}`   `    ``static` `void` `Backtrack(``int` `start, ``int` `target,` `                          ``List<``int``> factors,` `                          ``List > combinations,` `                          ``int` `n)` `    ``{` `        ``if` `(target == 1) {` `            ``// add a copy of the factors list to` `            ``// combinations, except if the list contains` `            ``// only one factor equal to n` `            ``if` `(factors.Count > 1 || factors[0] != n) {` `                ``combinations.Add(``new` `List<``int``>(factors));` `            ``}` `            ``return``;` `        ``}`   `        ``// try all factors from start to target, inclusive` `        ``for` `(``int` `i = start; i <= target; i++) {` `            ``if` `(target % i == 0) {` `                ``factors.Add(i);` `                ``Backtrack(i, target / i, factors,` `                          ``combinations, n);` `                ``factors.RemoveAt(factors.Count - 1);` `            ``}` `        ``}` `    ``}` `}` `}`

## Javascript

 `// Function to find all factor combinations of a number` `function` `backtrack(start, target, factors, combinations, n) {` `    ``// Base case: if target is 1, we have found a factorization` `    ``if` `(target === 1) {` `        ``// Add a copy of the factors array to combinations, except if the array contains only one factor equal to n` `        ``if` `(factors.length > 1 || factors[0] !== n) {` `            ``combinations.push([...factors]);` `        ``}` `        ``return``;` `    ``}`   `    ``// Try all factors from start to target, inclusive` `    ``for` `(let i = start; i <= target; i++) {` `        ``if` `(target % i === 0) {` `            ``factors.push(i);  ``// Add i to factors` `            ``backtrack(i, target / i, factors, combinations, n);  ``// Recursively find factors of target / i` `            ``factors.pop();  ``// Remove i from factors` `        ``}` `    ``}` `}`   `// Function to call backtrack and return the factor combinations` `function` `factorCombinations(n) {` `    ``const combinations = [];` `    ``const factors = [];` `    ``backtrack(2, n, factors, combinations, n);` `    ``return` `combinations;` `}`   `// Main code` `const n = 12;` `const combinations = factorCombinations(n);`   `console.log(`All the combinations of factors of \${n} are:`);` `for` `(let combination of combinations) {` `    ``console.log(combination.join(``' '``));` `}`

Output

```All the combinations of factors of 12 are:
[2, 2, 3]
[2, 6]
[3, 4]```

Time Complexity:  O(sqrt(n) * log(n)), where n is the input integer.
Auxiliary Space: O(log(n)), where n is the input integer.

#### Another Approach:

The code below is pure recursive code for printing all combinations of factors:

It uses a vector of integer to store a single list of factors and a vector of integer to store all combinations of factors. Instead of using an iterative loop, it uses the same recursive function to calculate all factor combinations.

## C++

 `// C++ program to print all factors combination` `#include ` `using` `namespace` `std;`   `// vector of vector for storing` `// list of factor combinations` `vector > factors_combination;`   `// recursive function` `void` `compute_factors(``int` `current_no, ``int` `n, ``int` `product,` `                     ``vector<``int``> single_list)` `{` `    `  `    ``// base case: if the product ` `    ``// exceeds our given number;` `    ``// OR` `    ``// current_no exceeds half the given n` `    ``if` `(current_no > (n / 2) || product > n)` `        ``return``;`   `    ``// if current list of factors` `    ``// is contributing to n` `    ``if` `(product == n) {` `      `  `        ``// storing the list` `        ``factors_combination.push_back(single_list); ` `      `  `        ``// into factors_combination` `        ``return``; ` `    ``}`   `    ``// including current_no in our list` `    ``single_list.push_back(current_no); `   `    ``// trying to get required ` `    ``// n with including current` `    ``// current_no` `    ``compute_factors(current_no, n, product * current_no,` `                    ``single_list);`   `    ``// excluding current_no from our list` `    ``single_list.pop_back(); `   `    ``// trying to get required n ` `    ``// without including current` `    ``// current_no` `    ``compute_factors(current_no + 1, n, product,` `                    ``single_list);` `}`   `// Driver Code` `int` `main()` `{` `    ``int` `n = 16;`   `    ``// vector to store single list of factors` `    ``// eg. 2,2,2,2 is one of the list for n=16` `    ``vector<``int``> single_list;`   `    ``// compute_factors ( starting_no, given_n,` `    ``// our_current_product, vector )` `    ``compute_factors(2, n, 1, single_list);`   `    ``// printing all possible factors stored in` `    ``// factors_combination` `    ``for` `(``int` `i = 0; i < factors_combination.size(); i++) {` `        ``for` `(``int` `j = 0; j < factors_combination[i].size();` `             ``j++)` `            ``cout << factors_combination[i][j] << ``" "``;` `        ``cout << endl;` `    ``}` `    ``return` `0;` `}`   `// code contributed by Devendra Kolhe`

## Java

 `// Java program to print all factors combination` `import` `java.util.*;`   `class` `GFG{` `    `  `// vector of vector for storing` `// list of factor combinations` `static` `Vector> factors_combination = ` `   ``new` `Vector>();` ` `  `// Recursive function` `static` `void` `compute_factors(``int` `current_no, ``int` `n, ``int` `product,` `                            ``Vector single_list)` `{` `      `  `    ``// base case: if the product` `    ``// exceeds our given number;` `    ``// OR` `    ``// current_no exceeds half the given n` `    ``if` `(current_no > (n / ``2``) || product > n)` `        ``return``;` `  `  `    ``// If current list of factors` `    ``// is contributing to n` `    ``if` `(product == n) ` `    ``{` `        `  `        ``// Storing the list` `        ``factors_combination.add(single_list);` `        `  `        ``// Printing all possible factors stored in` `        ``// factors_combination` `        ``for``(``int` `i = ``0``; i < factors_combination.size(); i++)` `        ``{` `            ``for``(``int` `j = ``0``; j < factors_combination.get(i).size(); j++)` `                ``System.out.print(factors_combination.get(i).get(j) + ``" "``);` `        ``}` `        ``System.out.println();` `        ``factors_combination = ``new` `Vector>();` `        `  `        ``// Into factors_combination` `        ``return``;` `    ``}` `  `  `    ``// Including current_no in our list` `    ``single_list.add(current_no);` `  `  `    ``// Trying to get required` `    ``// n with including current` `    ``// current_no` `    ``compute_factors(current_no, n, ` `                    ``product * current_no,` `                    ``single_list);` `  `  `    ``// Excluding current_no from our list` `    ``single_list.remove(single_list.size() - ``1``);` `  `  `    ``// Trying to get required n` `    ``// without including current` `    ``// current_no` `    ``compute_factors(current_no + ``1``, n, product, ` `                    ``single_list);` `}`   `// Driver code` `public` `static` `void` `main(String[] args)` `{` `    ``int` `n = ``16``;`   `    ``// Vector to store single list of factors` `    ``// eg. 2,2,2,2 is one of the list for n=16` `    ``Vector single_list = ``new` `Vector();` `  `  `    ``// compute_factors ( starting_no, given_n,` `    ``// our_current_product, vector )` `    ``compute_factors(``2``, n, ``1``, single_list);` `}` `}`   `// This code is contributed by decode2207`

## Python3

 `# Python3 program to print all factors combination`   `# vector of vector for storing` `# list of factor combinations` `factors_combination ``=` `[]`   `# recursive function` `def` `compute_factors(current_no, n, product, single_list):` `    ``global` `factors_combination` `    `  `    ``# base case: if the product` `    ``# exceeds our given number;` `    ``# OR` `    ``# current_no exceeds half the given n` `    ``if` `((current_no > ``int``(n ``/` `2``)) ``or` `(product > n)):` `        ``return` ` `  `    ``# if current list of factors` `    ``# is contributing to n` `    ``if` `(product ``=``=` `n):` `        ``# storing the list` `        ``factors_combination.append(single_list)` `        `  `        ``# printing all possible factors stored in` `        ``# factors_combination` `        ``for` `i ``in` `range``(``len``(factors_combination)):` `            ``for` `j ``in` `range``(``len``(factors_combination[i])):` `                ``print``(factors_combination[i][j], end``=``" "``)` `        ``print``()` `        ``factors_combination ``=` `[]` `        ``# into factors_combination` `        ``return` ` `  `    ``# including current_no in our list` `    ``single_list.append(current_no)` ` `  `    ``# trying to get required` `    ``# n with including current` `    ``# current_no` `    ``compute_factors(current_no, n, product ``*` `current_no, single_list)` ` `  `    ``# excluding current_no from our list` `    ``single_list.pop()` ` `  `    ``# trying to get required n` `    ``# without including current` `    ``# current_no` `    ``compute_factors(current_no ``+` `1``, n, product, single_list)`   `n ``=` `16`   `# vector to store single list of factors` `# eg. 2,2,2,2 is one of the list for n=16` `single_list ``=` `[]`   `# compute_factors ( starting_no, given_n,` `# our_current_product, vector )` `compute_factors(``2``, n, ``1``, single_list)`   `# This code is contributed by ukasp.`

## C#

 `// C# program to print all factors combination` `using` `System;` `using` `System.Collections.Generic;` `class` `GFG {` `    `  `    ``// vector of vector for storing` `    ``// list of factor combinations` `    ``static` `List> factors_combination = ``new` `List>();` `    `  `    ``// recursive function` `    ``static` `void` `compute_factors(``int` `current_no, ``int` `n, ``int` `product, List<``int``> single_list)` `    ``{` `         `  `        ``// base case: if the product` `        ``// exceeds our given number;` `        ``// OR` `        ``// current_no exceeds half the given n` `        ``if` `(current_no > (n / 2) || product > n)` `            ``return``;` `     `  `        ``// if current list of factors` `        ``// is contributing to n` `        ``if` `(product == n) {` `            `  `            ``// storing the list` `            ``factors_combination.Add(single_list);` `            ``// printing all possible factors stored in` `            ``// factors_combination` `            ``for``(``int` `i = 0; i < factors_combination.Count; i++)` `            ``{` `                ``for``(``int` `j = 0; j < factors_combination[i].Count; j++)` `                ``Console.Write(factors_combination[i][j] + ``" "``);` `            ``}` `            ``Console.WriteLine();` `            ``factors_combination = ``new` `List>();` `            ``// into factors_combination` `            ``return``;` `        ``}` `     `  `        ``// including current_no in our list` `        ``single_list.Add(current_no);` `     `  `        ``// trying to get required` `        ``// n with including current` `        ``// current_no` `        ``compute_factors(current_no, n, product * current_no, single_list);` `     `  `        ``// excluding current_no from our list` `        ``single_list.RemoveAt(single_list.Count - 1);` `     `  `        ``// trying to get required n` `        ``// without including current` `        ``// current_no` `        ``compute_factors(current_no + 1, n, product, single_list);` `    ``}`   `  ``static` `void` `Main() {` `    ``int` `n = 16;` ` `  `    ``// vector to store single list of factors` `    ``// eg. 2,2,2,2 is one of the list for n=16` `    ``List<``int``> single_list = ``new` `List<``int``>();` ` `  `    ``// compute_factors ( starting_no, given_n,` `    ``// our_current_product, vector )` `    ``compute_factors(2, n, 1, single_list);` `  ``}` `}`   `// This code is contributed by divyesh072019.`

## Javascript

 ``

Output

```2 2 2 2
2 2 4
2 8
4 4 ```

Time Complexity: O(n2) , n is the size of vector
Auxiliary Space: O(n2), n is the size of vector

Please suggest if someone has a better solution which is more efficient in terms of space and time.