# Check if an array can be divided into pairs whose sum is divisible by k

Given an array of integers and a number k, write a function that returns true if given array can be divided into pairs such that sum of every pair is divisible by k.

Examples:

```Input: arr[] = {9, 7, 5, 3},
k = 6
Output: True
We can divide array into (9, 3) and
(7, 5). Sum of both of these pairs
is a multiple of 6.

Input: arr[] = {92, 75, 65, 48, 45, 35},
k = 10
Output: True
We can divide array into (92, 48), (75, 65)
and (45, 35). Sum of all these pairs is a
multiple of 10.

Input: arr[] = {91, 74, 66, 48}, k = 10
Output: False
```

A Simple Solution is to iterate through every element arr[i]. Find if there is another not yet visited element that has remainder as (k – arr[i]%k). If there is no such element, return false. If a pair is found, then mark both elements as visited. Time complexity of this solution is O(n2 and it requires O(n) extra space.

An Efficient Solution is to use Hashing.

```1) If length of given array is odd, return false.
An odd length array cannot be divided in pairs.
2) Traverse input array and count occurrences of
all remainders.
freq[arr[i] % k]++
3) Traverse input array again.
a) Find remainder of current element.
b) If remainder divides k into two halves, then
there must be even occurrences of it as it
forms  pair with itself only.
c) If remainder is 0, then then there must be
even occurrences.
c) Else, number of occurrences of current
remainder must be equal to number of
occurrences of "k - current remainder".
```

Time complexity of above algorithm is O(n).

Below implementation uses map in C++ STL. The map is typically implemented using Red-Black Tree and takes O(Log n) time for access. Therefore time complexity of below implementation is O(n Log n), but the algorithm can be easily implemented in O(n) time using hash table.

## C++

 `// A C++ program to check if arr[0..n-1] can be divided ` `// in pairs such that every pair is divisible by k. ` `#include ` `using` `namespace` `std; ` ` `  `// Returns true if arr[0..n-1] can be divided into pairs ` `// with sum divisible by k. ` `bool` `canPairs(``int` `arr[], ``int` `n, ``int` `k) ` `{ ` `    ``// An odd length array cannot be divided into pairs ` `    ``if` `(n & 1) ` `        ``return` `false``; ` ` `  `    ``// Create a frequency array to count occurrences ` `    ``// of all remainders when divided by k. ` `    ``map<``int``, ``int``> freq; ` ` `  `    ``// Count occurrences of all remainders ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``freq[arr[i] % k]++; ` ` `  `    ``// Traverse input array and use freq[] to decide ` `    ``// if given array can be divided in pairs ` `    ``for` `(``int` `i = 0; i < n; i++) ` `    ``{ ` `        ``// Remainder of current element ` `        ``int` `rem = arr[i] % k; ` ` `  `        ``// If remainder with current element divides ` `        ``// k into two halves. ` `        ``if`  `(2*rem == k) ` `        ``{ ` `            ``// Then there must be even occurrences of ` `            ``// such remainder ` `            ``if` `(freq[rem] % 2 != 0) ` `                ``return` `false``; ` `        ``} ` ` `  `        ``// If remainder is 0, then there must be two  ` `        ``// elements with 0 remainder ` `        ``else` `if` `(rem == 0) ` `        ``{ ` `           ``if` `(freq[rem] & 1)            ` `               ``return` `false``; ` `        ``}         ` ` `  `        ``// Else number of occurrences of remainder ` `        ``// must be equal to number of occurrences of ` `        ``// k - remainder ` `        ``else` `if` `(freq[rem] != freq[k - rem]) ` `            ``return` `false``; ` `    ``} ` `    ``return` `true``; ` `} ` ` `  `/* Driver program to test above function */` `int` `main() ` `{ ` `    ``int` `arr[] =  {92, 75, 65, 48, 45, 35}; ` `    ``int` `k = 10; ` `    ``int` `n = ``sizeof``(arr)/``sizeof``(arr[0]); ` `    ``canPairs(arr, n, k)? cout << ``"True"``: cout << ``"False"``; ` `    ``return` `0; ` `} `

## Java

 `  `  `import` `java.util.HashMap; ` ` `  `public` `class` `Divisiblepair  ` `{ ` `    ``// Returns true if arr[0..n-1] can be divided into pairs ` `    ``// with sum divisible by k. ` `    ``static` `boolean` `canPairs(``int` `ar[], ``int` `k)  ` `    ``{ ` `        ``// An odd length array cannot be divided into pairs ` `        ``if` `(ar.length % ``2` `== ``1``) ` `            ``return` `false``; ` `         `  `        ``// Create a frequency array to count occurrences ` `        ``// of all remainders when divided by k. ` `        ``HashMap hm = ``new` `HashMap<>(); ` `         `  `        ``// Count occurrences of all remainders ` `        ``for` `(``int` `i = ``0``; i < ar.length; i++)  ` `        ``{ ` `            ``int` `rem = ar[i] % k; ` `            ``if` `(!hm.containsKey(rem))  ` `            ``{ ` `                ``hm.put(rem, ``0``); ` `            ``} ` `            ``hm.put(rem, hm.get(rem) + ``1``); ` `        ``} ` `         `  `        ``// Traverse input array and use freq[] to decide ` `        ``// if given array can be divided in pairs ` `        ``for` `(``int` `i = ``0``; i < ar.length; i++)  ` `        ``{ ` `             ``// Remainder of current element ` `            ``int` `rem = ar[i] % k; ` `             `  `            ``// If remainder with current element divides ` `            ``// k into two halves. ` `            ``if` `(``2` `* rem == k)  ` `            ``{ ` `                ``// Then there must be even occurrences of ` `                ``// such remainder ` `                ``if` `(hm.get(rem) % ``2` `== ``1``) ` `                    ``return` `false``; ` `            ``}  ` `             `  `            ``// If remainder is 0, then there must be two  ` `            ``// elements with 0 remainder ` `            ``else` `if` `(rem == ``0``)  ` `            ``{ ` `                ``// Then there must be even occurrences of ` `                ``// such remainder ` `                ``if` `(hm.get(rem) % ``2` `== ``1``) ` `                    ``return` `false``; ` `            ``} ` `             `  `            ``// Else number of occurrences of remainder ` `            ``// must be equal to number of occurrences of ` `            ``// k - remainder ` `            ``else`  `            ``{ ` `                ``if` `(hm.get(k - rem) != hm.get(rem)) ` `                    ``return` `false``; ` `            ``} ` `        ``} ` `        ``return` `true``; ` `    ``} ` ` `  `    ``// Driver program to test above functions ` `    ``public` `static` `void` `main(String[] args)  ` `    ``{ ` `        ``int` `arr[] = { ``92``, ``75``, ``65``, ``48``, ``45``, ``35` `}; ` `        ``int` `k = ``10``; ` `        ``boolean` `ans = canPairs(arr, k); ` `        ``if` `(ans) ` `            ``System.out.println(``"True"``); ` `        ``else` `            ``System.out.println(``"False"``); ` ` `  `    ``} ` `} ` ` `  `// This code is contributed by Rishabh Mahrsee `

Output:

`True`