# Count pairs in Array whose product is divisible by K

• Difficulty Level : Hard
• Last Updated : 24 Mar, 2022

Given a array vec and an integer K, count the number of pairs (i, j) such that vec[i]*vec[j] is divisible by K where i<j.

Examples:

Input: vec = {1, 2, 3, 4, 5, 6}, K = 4
Output: 6
Explanation: The pairs of indices (0, 3), (1, 3), (2, 3), (3, 4), (3, 5) and (1, 5) satisfy the condition as their products respectively are 4, 8, 12, 20, 24, 12
which are divisible by k=4.
Since there are 6 pairs the count will be 6.

Input: vec = {1, 2, 3, 4}, K = 2
Output: 5
Explanation: The pairs of indices (0, 1), (1, 2), (1, 3), (0, 3), (2, 3) satisfy the condition as their products respectively are 2, 6, 8, 4, 12 which are divisible by k=2.
Since there are 5 pairs the count will be 5.

Naive Approach: The most simple approach to solve this problem is to find all pairs, and for each pair, check if their product is divisible by K.

Time Complexity: O(N^2)
Auxiliary space: O(1)

Efficient Approach: The above problem can be solved efficiently with the help of GCD.

We know that product of A and B is divisible by a number B if GCD(A, B) = B.

Similarly (vec[i] * vec[j]) will be divisible by K if their GCD is K.

Follow the below steps to solve the problem:

• Create a countarr of size 10^5 + 1 to precalculate the count of how many numbers are divisible by a number num(say) (for all num 1 to 10^5).
• Then we just need to loop for each element or vector and finding out the remaining number( remaining_factor) needed to be multiplied to it so as to make the GCD equals to k.
• Then for this number the count of pairs will be as much as the multiple of remaining_factor present in array (ignoring the number itself).
• Thus adding count of pairs for each i in vec we will get our answer, we will  divide the final answer by 2 as we have counted each pair twice for any pair (i, j) and remove the duplicates.

Below is the implementation of the above approach:

## C++

 `// C++ program for Count number of pairs` `// in a vector such that` `// product is divisible by K`   `#include ` `using` `namespace` `std;`   `// Precalculate count array to see numbers` `// that are divisible by a number num (say)` `// for all num 1 to 10^5` `int` `countarr;` `int` `count_of_multiples;`   `long` `long` `countPairs(vector<``int``>& vec, ``int` `k)` `{` `    ``int` `n = vec.size();` `    ``for` `(``int` `i = 0; i < n; i++)`   `        ``// counting frequency of  each` `        ``// element in vector` `        ``countarr[vec[i]]++;`   `    ``for` `(``int` `i = 1; i < 100001; i++) {` `        ``for` `(``int` `j = i; j < 100001; j = j + i)`   `            ``// counting total elements present in` `            ``// array which are multiple of i` `            ``count_of_multiples[i] += countarr[j];` `    ``}`   `    ``long` `long` `ans = 0;` `    ``for` `(``int` `i = 0; i < n; i++) {` `        ``long` `long` `factor = __gcd(k, vec[i]);` `        ``long` `long` `remaining_factor = (k / factor);` `        ``long` `long` `j = count_of_multiples[remaining_factor];`   `        ``// if vec[i] itself is multiple of` `        ``// remaining factor then we to ignore` `        ``// it as  i!=j` `        ``if` `(vec[i] % remaining_factor == 0)` `            ``j--;` `        ``ans += j;` `    ``}`   `    ``// as we have counted any distinct pair` `    ``// (i, j) two times, we need to take them` `    ``// only once` `    ``ans /= 2;`   `    ``return` `ans;` `}`   `// Driver code` `int` `main()` `{` `    ``vector<``int``> vec = { 1, 2, 3, 4, 5, 6 };` `    ``int` `k = 4;` `    ``cout << countPairs(vec, k) << endl;` `}`

## Java

 `// Java program for Count number of pairs` `// in a vector such that` `// product is divisible by K` `import` `java.util.*;` `public` `class` `GFG {`   `  ``// Precalculate count array to see numbers` `  ``// that are divisible by a number num (say)` `  ``// for all num 1 to 10^5` `  ``static` `int``[] countarr = ``new` `int``[``100001``];` `  ``static` `int``[] count_of_multiples = ``new` `int``[``100001``];`   `  ``// Recursive function to return` `  ``// gcd of a and b` `  ``static` `int` `gcd(``int` `a, ``int` `b)` `  ``{` `    ``if` `(b == ``0``)` `      ``return` `a;` `    ``return` `gcd(b, a % b);` `  ``}`   `  ``static` `long` `countPairs(``int``[] vec, ``int` `k)` `  ``{` `    ``int` `n = vec.length;` `    ``for` `(``int` `i = ``0``; i < n; i++)`   `      ``// counting frequency of  each` `      ``// element in vector` `      ``countarr[vec[i]]++;`   `    ``for` `(``int` `i = ``1``; i < ``100001``; i++) {` `      ``for` `(``int` `j = i; j < ``100001``; j = j + i)`   `        ``// counting total elements present in` `        ``// array which are multiple of i` `        ``count_of_multiples[i] += countarr[j];` `    ``}`   `    ``long` `ans = ``0``;` `    ``for` `(``int` `i = ``0``; i < n; i++) {` `      ``long` `factor = gcd(k, vec[i]);` `      ``long` `remaining_factor = (k / factor);` `      ``long` `j = count_of_multiples[(``int``)remaining_factor];`   `      ``// if vec[i] itself is multiple of` `      ``// remaining factor then we to ignore` `      ``// it as  i!=j` `      ``if` `(vec[i] % remaining_factor == ``0``)` `        ``j--;` `      ``ans += j;` `    ``}`   `    ``// as we have counted any distinct pair` `    ``// (i, j) two times, we need to take them` `    ``// only once` `    ``ans /= ``2``;`   `    ``return` `ans;` `  ``}`   `  ``// Driver code` `  ``public` `static` `void` `main(String args[])` `  ``{` `    ``int``[] vec = { ``1``, ``2``, ``3``, ``4``, ``5``, ``6` `};` `    ``int` `k = ``4``;` `    ``System.out.println(countPairs(vec, k));` `  ``}` `}`   `// This code is contributed by Samim Hossain Mondal.`

## Python3

 `# Python program for Count number of pairs` `# in a vector such that` `# product is divisible by K` `import` `math`   `# Precalculate count array to see numbers` `# that are divisible by a number num (say)` `# for all num 1 to 10^5` `countarr ``=` `[``0``] ``*` `100001` `count_of_multiples ``=` `[``0``] ``*` `100001`   `def` `countPairs(vec, k):`   `    ``n ``=` `len``(vec)` `    ``for` `i ``in` `range``(``0``, n):`   `        ``# counting frequency of  each` `        ``# element in vector` `        ``countarr[vec[i]] ``+``=` `1`   `    ``for` `i ``in` `range``(``1``, ``100001``):` `        ``j ``=` `i` `        ``while``(j < ``100001``):`   `            ``# counting total elements present in` `            ``# array which are multiple of i` `            ``count_of_multiples[i] ``+``=` `countarr[j]` `            ``j ``+``=` `i`   `    ``ans ``=` `0` `    ``for` `i ``in` `range``(``0``, n):` `        ``factor ``=` `math.gcd(k, vec[i])` `        ``remaining_factor ``=` `(k ``/``/` `factor)` `        ``j ``=` `count_of_multiples[remaining_factor]`   `        ``# if vec[i] itself is multiple of` `        ``# remaining factor then we to ignore` `        ``# it as  i!=j` `        ``if` `(vec[i] ``%` `remaining_factor ``=``=` `0``):` `            ``j ``-``=` `1` `        ``ans ``+``=` `j`   `    ``# as we have counted any distinct pair` `    ``# (i, j) two times, we need to take them` `    ``# only once` `    ``ans ``/``/``=` `2` `    ``return` `ans`   `# Driver code` `vec ``=` `[``1``, ``2``, ``3``, ``4``, ``5``, ``6``]` `k ``=` `4` `print``(countPairs(vec, k))`   `# This code is contributed by Samim Hossain Mondal.`

## C#

 `// C# program for Count number of pairs` `// in a vector such that` `// product is divisible by K`   `using` `System;` `class` `GFG {`   `  ``// Precalculate count array to see numbers` `  ``// that are divisible by a number num (say)` `  ``// for all num 1 to 10^5` `  ``static` `int``[] countarr = ``new` `int``;` `  ``static` `int``[] count_of_multiples = ``new` `int``;`   `  ``// Recursive function to return` `  ``// gcd of a and b` `  ``static` `int` `gcd(``int` `a, ``int` `b)` `  ``{` `    ``if` `(b == 0)` `      ``return` `a;` `    ``return` `gcd(b, a % b);` `  ``}`   `  ``static` `long` `countPairs(``int``[] vec, ``int` `k)` `  ``{` `    ``int` `n = vec.Length;` `    ``for` `(``int` `i = 0; i < n; i++)`   `      ``// counting frequency of  each` `      ``// element in vector` `      ``countarr[vec[i]]++;`   `    ``for` `(``int` `i = 1; i < 100001; i++) {` `      ``for` `(``int` `j = i; j < 100001; j = j + i)`   `        ``// counting total elements present in` `        ``// array which are multiple of i` `        ``count_of_multiples[i] += countarr[j];` `    ``}`   `    ``long` `ans = 0;` `    ``for` `(``int` `i = 0; i < n; i++) {` `      ``long` `factor = gcd(k, vec[i]);` `      ``long` `remaining_factor = (k / factor);` `      ``long` `j = count_of_multiples[remaining_factor];`   `      ``// if vec[i] itself is multiple of` `      ``// remaining factor then we to ignore` `      ``// it as  i!=j` `      ``if` `(vec[i] % remaining_factor == 0)` `        ``j--;` `      ``ans += j;` `    ``}`   `    ``// as we have counted any distinct pair` `    ``// (i, j) two times, we need to take them` `    ``// only once` `    ``ans /= 2;`   `    ``return` `ans;` `  ``}`   `  ``// Driver code` `  ``public` `static` `void` `Main()` `  ``{` `    ``int``[] vec = { 1, 2, 3, 4, 5, 6 };` `    ``int` `k = 4;` `    ``Console.WriteLine(countPairs(vec, k));` `  ``}` `}`   `// This code is contributed by Samim Hossain Mondal.`

## Javascript

 ``

Output:

`6`

Time Complexity: O(N*log(N))
Auxiliary Space: O(N)

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