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Number of pairs such that their HCF and LCM is equal
• Last Updated : 04 May, 2020

Given an array a[] of non-negative integers. The task is to count the number of pairs (i, j) in the array such that LCM(a[i], a[j]) = HCF(a[i], a[j]).

Note: The pair (i, j) and (j, i) are considered same and i should not be equal to j.

Examples:

```Input : a[] = {3, 4, 3, 4, 5}
Output : 2
Pairs are (3, 3) and (4, 4)

Input  : a[] = {1, 1, 1}
Output : 3
```

## Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

Naive approach: Generate all possible pairs and count pairs with equal HCF and LCM.

## C++

 `// Naive C++ program to count number of pairs ` `// such that their hcf and lcm are equal ` `#include ` `using` `namespace` `std; ` ` `  `// Function to return HCF of two numbers ` `int` `gcd(``int` `a, ``int` `b) ` `{ ` `    ``if` `(a == 0) ` `        ``return` `b; ` `    ``return` `gcd(b % a, a); ` `} ` ` `  `// Function to return LCM of two numbers ` `int` `lcm(``int` `a, ``int` `b) ` `{ ` `    ``return` `(a * b) / gcd(a, b); ` `} ` ` `  `// Returns the number of valid pairs ` `int` `countPairs(``int` `arr[], ``int` `n) ` `{ ` `    ``int` `ans = 0; ``// initializing answer ` ` `  `    ``// Traversing the array. For each array ` `    ``// element, checking if it ` `    ``// follow the condition ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``for` `(``int` `j = i + 1; j < n; j++) ` `            ``if` `(lcm(arr[i], arr[j]) == gcd(arr[i], arr[j])) ` `                ``ans++; ` `    ``return` `ans; ` `} ` ` `  `// Driver function ` `int` `main() ` `{ ` `    ``int` `arr[] = { 1, 1, 1 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` `    ``cout << countPairs(arr, n); ` `    ``return` `0; ` `} `

## Java

 `// Naive Java program to count number of pairs ` `// such that their hcf and lcm are equal ` `import` `java.util.*; ` ` `  `class` `GFG{ ` `  `  `// Function to return HCF of two numbers ` `static` `int` `gcd(``int` `a, ``int` `b) ` `{ ` `    ``if` `(a == ``0``) ` `        ``return` `b; ` `    ``return` `gcd(b % a, a); ` `} ` `  `  `// Function to return LCM of two numbers ` `static` `int` `lcm(``int` `a, ``int` `b) ` `{ ` `    ``return` `(a * b) / gcd(a, b); ` `} ` `  `  `// Returns the number of valid pairs ` `static` `int` `countPairs(``int` `arr[], ``int` `n) ` `{ ` `    ``int` `ans = ``0``; ``// initializing answer ` `  `  `    ``// Traversing the array. For each array ` `    ``// element, checking if it ` `    ``// follow the condition ` `    ``for` `(``int` `i = ``0``; i < n; i++) ` `        ``for` `(``int` `j = i + ``1``; j < n; j++) ` `            ``if` `(lcm(arr[i], arr[j]) == gcd(arr[i], arr[j])) ` `                ``ans++; ` `    ``return` `ans; ` `} ` `  `  `// Driver function ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `arr[] = { ``1``, ``1``, ``1` `}; ` `    ``int` `n = arr.length; ` `    ``System.out.print(countPairs(arr, n)); ` `} ` `} ` ` `  `// This code is contributed by Rajput-Ji `

## Python3

 `# Naive Python program to count number of pairs ` `# such that their hcf and lcm are equal ` ` `  `# Function to return HCF of two numbers ` `def` `gcd(a, b): ` `    ``if` `(a ``=``=` `0``): ` `        ``return` `b; ` `    ``return` `gcd(b ``%` `a, a); ` ` `  `# Function to return LCM of two numbers ` `def` `lcm(a, b): ` `    ``return` `(a ``*` `b) ``/` `gcd(a, b); ` ` `  `# Returns the number of valid pairs ` `def` `countPairs(arr, n): ` `    ``ans ``=` `0``; ``# initializing answer ` ` `  `    ``# Traversing the array. For each array ` `    ``# element, checking if it ` `    ``# follow the condition ` `    ``for` `i ``in` `range``(n): ` `        ``for` `j ``in` `range``(i``+``1``,n): ` `            ``if` `(lcm(arr[i], arr[j]) ``=``=` `gcd(arr[i], arr[j])): ` `                ``ans``+``=``1``; ` `    ``return` `ans; ` ` `  `# Driver function ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``arr ``=` `[ ``1``, ``1``, ``1` `]; ` `    ``n ``=` `len``(arr); ` `    ``print``(countPairs(arr, n)); ` ` `  `# This code is contributed by 29AjayKumar `

## C#

 `// Naive C# program to count number of pairs ` `// such that their hcf and lcm are equal ` `using` `System; ` ` `  `class` `GFG{ ` `   `  `// Function to return HCF of two numbers ` `static` `int` `gcd(``int` `a, ``int` `b) ` `{ ` `    ``if` `(a == 0) ` `        ``return` `b; ` `    ``return` `gcd(b % a, a); ` `} ` `   `  `// Function to return LCM of two numbers ` `static` `int` `lcm(``int` `a, ``int` `b) ` `{ ` `    ``return` `(a * b) / gcd(a, b); ` `} ` `   `  `// Returns the number of valid pairs ` `static` `int` `countPairs(``int` `[]arr, ``int` `n) ` `{ ` `    ``int` `ans = 0; ``// initializing answer ` `   `  `    ``// Traversing the array. For each array ` `    ``// element, checking if it ` `    ``// follow the condition ` `    ``for` `(``int` `i = 0; i < n; i++) ` `        ``for` `(``int` `j = i + 1; j < n; j++) ` `            ``if` `(lcm(arr[i], arr[j]) == gcd(arr[i], arr[j])) ` `                ``ans++; ` `    ``return` `ans; ` `} ` `   `  `// Driver function ` `public` `static` `void` `Main(String[] args) ` `{ ` `    ``int` `[]arr = { 1, 1, 1 }; ` `    ``int` `n = arr.Length; ` `    ``Console.Write(countPairs(arr, n)); ` `} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

Output:

```3
```

Efficient Approach: If we observe carefully, it can be proved that the HCF and LCM of two numbers can be equal only when the numbers are also equal.

Proof:

```Let,
HCF(a[i], a[j]) = LCM(a[i], a[j]) = K

Since HCF(a[i], a[j]) = k,
a[i] = k*n1, a[j] = k*n2, for some natural numbers n1, n2

We know that,
HCF × LCM = Product of the two numbers

Therefore,
k*k = k*n1 × k*n2
or, n1*n2 = 1

Implies, n1 = n2 = 1, since n1, n2 are natural numbers.

Therefore,
a[i] = k*n1 = k, a[j] = k*n2 = k
i.e. the numbers must be equal.
```

So we have to count pairs wih same elements in the pair.

It is observed that only the pairs of the form (arr[i], arr[j]) where arr[i] = arr[j] will satisfy the given condition. So, the problem now gets reduced to finding the number of pairs (arr[i], arr[j]) such that arr[i] = arr[j].

Below is the implementation of the above approach:

## C++

 `  `  ` `  `// C++ program to count number of pairs ` `// such that their hcf and lcm are equal ` `#include ` `using` `namespace` `std; ` ` `  `// Function to count number of pairs ` `// such that their hcf and lcm are equal ` `int` `countPairs(``int` `a[], ``int` `n) ` `{ ` `    ``// Store frequencies of array elements ` `    ``unordered_map<``int``, ``int``> frequency; ` `    ``for` `(``int` `i = 0; i < n; i++) { ` `        ``frequency[a[i]]++; ` `    ``} ` ` `  `    ``int` `count = 0; ` ` `  `    ``// Count of pairs (arr[i], arr[j]) ` `    ``// where arr[i] = arr[j] ` `    ``for` `(``auto` `x : frequency) { ` `        ``int` `f = x.second; ` `        ``count += f * (f - 1) / 2; ` `    ``} ` ` `  `    ``// Count of pairs (arr[i], arr[j]) where ` `    ``// arr[i] = arr[j], ` `    ``return` `count; ` `} ` ` `  `// Driver function ` `int` `main() ` `{ ` `    ``int` `arr[] = { 1, 1, 1 }; ` `    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr); ` `    ``cout << countPairs(arr, n); ` `    ``return` `0; ` `} `

## Java

 `// Java program to count number of pairs ` `// such that their hcf and lcm are equal ` ` `  `import` `java.util.*; ` ` `  `class` `GFG{ ` `  `  `// Function to count number of pairs ` `// such that their hcf and lcm are equal ` `static` `int` `countPairs(``int` `arr[], ``int` `n) ` `{ ` `    ``// Store frequencies of array elements ` `    ``HashMap frequency = ` `            ``new` `HashMap(); ` `    ``for` `(``int` `i = ``0``; i < n; i++) { ` `        ``if``(frequency.containsKey(arr[i])){ ` `            ``frequency.put(arr[i], frequency.get(arr[i])+``1``); ` `        ``}``else``{ ` `            ``frequency.put(arr[i], ``1``); ` `    ``} ` `    ``} ` `  `  `    ``int` `count = ``0``; ` `  `  `    ``// Count of pairs (arr[i], arr[j]) ` `    ``// where arr[i] = arr[j] ` `    ``for` `(Map.Entry x : frequency.entrySet()) { ` `        ``int` `f = x.getValue(); ` `        ``count += f * (f - ``1``) / ``2``; ` `    ``} ` `  `  `    ``// Count of pairs (arr[i], arr[j]) where ` `    ``// arr[i] = arr[j], ` `    ``return` `count; ` `} ` `  `  `// Driver function ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `arr[] = { ``1``, ``1``, ``1` `}; ` `    ``int` `n = arr.length; ` `    ``System.out.print(countPairs(arr, n)); ` `} ` `} ` ` `  `// This code contributed by sapnasingh4991 `

## C#

 `// C# program to count number of pairs ` `// such that their hcf and lcm are equal ` `using` `System; ` `using` `System.Collections.Generic; ` ` `  `class` `GFG{ ` ` `  `// Function to count number of pairs ` `// such that their hcf and lcm are equal ` `static` `int` `countPairs(``int` `[]a, ``int` `n) ` `{ ` `    ``// Store frequencies of array elements ` `    ``Dictionary<``int``, ``int``> frequency = ``new` `Dictionary<``int``,``int``>(); ` `        ``for` `(``int` `i = 0; i < n; i++)  ` `        ``{  ` `            ``if` `(frequency.ContainsKey(a[i]))   ` `            ``{  ` `                ``var` `val = frequency[a[i]];  ` `                ``frequency.Remove(a[i]);  ` `                ``frequency.Add(a[i], val + 1);   ` `            ``}   ` `            ``else` `            ``{  ` `                ``frequency.Add(a[i], 1);  ` `            ``}  ` `        ``}  ` `          `  `    ``int` `count = 0; ` ` `  `    ``// Count of pairs (arr[i], arr[j]) ` `    ``// where arr[i] = arr[j] ` `    ``foreach``(KeyValuePair<``int``, ``int``> entry ``in` `frequency) { ` `        ``int` `f = entry.Value; ` `        ``count += f * (f - 1) / 2; ` `    ``} ` ` `  `    ``// Count of pairs (arr[i], arr[j]) where ` `    ``// arr[i] = arr[j], ` `    ``return` `count; ` `} ` ` `  `// Driver function ` `public` `static` `void` `Main(String[] args) ` `{ ` `    ``int` `[]arr = { 1, 1, 1 }; ` `    ``int` `n = arr.Length; ` `    ``Console.Write(countPairs(arr, n)); ` `} ` `} ` ` `  `// This code is contributed by shivanisinghss2110 `

## Python3

 `# Python 3 program to count number of pairs ` `# such that their hcf and lcm are equal ` `from` `collections ``import` `defaultdict ` `  `  `# Function to count number of pairs ` `# such that their hcf and lcm are equal ` `def` `countPairs(a, n): ` ` `  `    ``# Store frequencies of array elements ` `    ``frequency ``=` `defaultdict(``int``) ` `    ``for` `i ``in` `range``(n) : ` `        ``frequency[a[i]] ``+``=` `1` `     `  `  `  `    ``count ``=` `0` `  `  `    ``# Count of pairs (arr[i], arr[j]) ` `    ``# where arr[i] = arr[j] ` `    ``for` `x ``in` `frequency.keys(): ` `        ``f ``=` `frequency[x] ` `        ``count ``+``=` `f ``*` `(f ``-` `1``) ``/``/` `2` `  `  `    ``# Count of pairs (arr[i], arr[j]) where ` `    ``# arr[i] = arr[j], ` `    ``return` `count ` `  `  `# Driver function ` `if` `__name__ ``=``=` `"__main__"``: ` `     `  `    ``arr ``=` `[ ``1``, ``1``, ``1` `] ` `    ``n ``=` `len``(arr) ` `    ``print``(countPairs(arr, n)) ` ` `  `# This code is contributed by chitranayal `

Output:

```3
```

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