# Count of pairs of (i, j) such that ((n % i) % j) % n is maximized

Given an integer n, the task is to count the number of pairs (i, j) such that ((n % i) % j) % n is maximized where 1 ≤ i, j ≤ n

Examples:

Input: n = 5
Output: 3
(3, 3), (3, 4) and (3, 5) are the only valid pairs.

Input: n = 55
Output: 28

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Naive Approach: To obtain the maximum remainder value, n has to be divided by (n / 2) + 1. Store max = n % ((n / 2) + 1), now check for all possible values of i and j. If ((n % i) % j) % n = max then update count = count + 1. Print the count in the end.

## C++

 `// CPP implementation of the approach ` `#include ` `using` `namespace` `std; ` ` `  `// Function to return the count of required pairs ` `int` `countPairs(``int` `n) ` `{ ` `    ``// Number which will give the max value ` `    ``// for ((n % i) % j) % n ` `    ``int` `num = ((n / 2) + 1); ` `     `  `    ``// To store the maximum possible value of ` `    ``// ((n % i) % j) % n ` `    ``int` `max = n % num; ` ` `  `    ``// To store the count of possible pairs ` `    ``int` `count = 0; ` ` `  `    ``// Check all possible pairs ` `    ``for` `(``int` `i = 1; i <= n; i++)  ` `    ``{ ` `        ``for` `(``int` `j = 1; j <= n; j++) ` `        ``{ ` ` `  `            ``// Calculating the value of ((n % i) % j) % n ` `            ``int` `val = ((n % i) % j) % n; ` ` `  `            ``// If value is equal to maximum ` `            ``if` `(val == max) ` `                ``count++; ` `        ``} ` `    ``} ` ` `  `    ``// Return the number of possible pairs ` `    ``return` `count; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `n = 5; ` `    ``cout << (countPairs(n)); ` `} ` ` `  `// This code is contributed by  ` `// Surendra_Gangwar `

## Java

 `// Java implementation of the approach ` `class` `GFG { ` ` `  `    ``// Function to return the count of required pairs ` `    ``public` `static` `int` `countPairs(``int` `n) ` `    ``{ ` `        ``// Number which will give the max value ` `        ``// for ((n % i) % j) % n ` `        ``int` `num = ((n / ``2``) + ``1``); ` ` `  `        ``// To store the maximum possible value of ` `        ``// ((n % i) % j) % n ` `        ``int` `max = n % num; ` ` `  `        ``// To store the count of possible pairs ` `        ``int` `count = ``0``; ` ` `  `        ``// Check all possible pairs ` `        ``for` `(``int` `i = ``1``; i <= n; i++) { ` `            ``for` `(``int` `j = ``1``; j <= n; j++) { ` ` `  `                ``// Calculating the value of ((n % i) % j) % n ` `                ``int` `val = ((n % i) % j) % n; ` ` `  `                ``// If value is equal to maximum ` `                ``if` `(val == max) ` `                    ``count++; ` `            ``} ` `        ``} ` ` `  `        ``// Return the number of possible pairs ` `        ``return` `count; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `n = ``5``; ` `        ``System.out.println(countPairs(n)); ` `    ``} ` `} `

## Python3

 `# Python3 implementation of the approach ` ` `  `# Function to return the count of  ` `# required pairs ` `def` `countPairs(n): ` ` `  `    ``# Number which will give the Max  ` `    ``# value for ((n % i) % j) % n ` `    ``num ``=` `((n ``/``/` `2``) ``+` `1``) ` `     `  `    ``# To store the Maximum possible value  ` `    ``# of ((n % i) % j) % n ` `    ``Max` `=` `n ``%` `num ` ` `  `    ``# To store the count of possible pairs ` `    ``count ``=` `0` ` `  `    ``# Check all possible pairs ` `    ``for` `i ``in` `range``(``1``, n ``+` `1``): ` `     `  `        ``for` `j ``in` `range``(``1``, n ``+` `1``): ` ` `  `            ``# Calculating the value of ` `            ``# ((n % i) % j) % n ` `            ``val ``=` `((n ``%` `i) ``%` `j) ``%` `n ` ` `  `            ``# If value is equal to Maximum ` `            ``if` `(val ``=``=` `Max``): ` `                ``count ``+``=` `1` `         `  `    ``# Return the number of possible pairs ` `    ``return` `count ` ` `  `# Driver code ` `n ``=` `5` `print``(countPairs(n)) ` ` `  `# This code is contributed ` `# by Mohit Kumar `

## C#

 `// C# implementation of the above approach  ` `using` `System; ` ` `  `class` `GFG ` `{  ` ` `  `// Function to return the count of required pairs  ` `static` `int` `countPairs(``int` `n)  ` `{  ` `    ``// Number which will give the max  ` `    ``// value for ((n % i) % j) % n  ` `    ``int` `num = ((n / 2) + 1) ; ` ` `  `    ``// To store the maximum possible value  ` `    ``// of ((n % i) % j) % n  ` `    ``int` `max = n % num;  ` ` `  `    ``// To store the count of possible pairs  ` `    ``int` `count = 0;  ` ` `  `    ``// Check all possible pairs  ` `    ``for` `(``int` `i = 1; i <= n; i++)  ` `    ``{  ` `        ``for` `(``int` `j = 1; j <= n; j++)  ` `        ``{  ` ` `  `            ``// Calculating the value of  ` `            ``// ((n % i) % j) % n  ` `            ``int` `val = ((n % i) % j) % n;  ` ` `  `            ``// If value is equal to maximum  ` `            ``if` `(val == max)  ` `                ``count++;  ` `        ``}  ` `    ``}  ` ` `  `    ``// Return the number of possible pairs  ` `    ``return` `count;  ` `}  ` ` `  `// Driver code  ` `public` `static` `void` `Main()  ` `{  ` `    ``int` `n = 5;  ` `    ``Console.WriteLine(countPairs(n));  ` `}  ` `} ` ` `  `// This code is contributed by Ryuga ` ` `

## PHP

 ` `

Output:

```3
```

Time Complexity: O(n2)

Efficient Approach: Get the maximum value for remainder i.e. max = n % num where num = ((n / 2) + 1). Now i has to be chosen as num in order to obtain the maximum value and j can be chosen as any value from the range [max, n] because we don’t need to reduce the maximum value calculated and choosing j > max will not affect the previous value obtained. So the total pairs will be n – max.
This approach will not work for n = 2. This is because, for n = 2, maximum remainder will be 0 and n – max will give 2 but we know that the answer is 4. All possible pairs in this case are (1, 1), (1, 2), (2, 1) and (2, 2).

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` `#include ` `using` `namespace` `std; ` ` `  `// Function to return the count of ` `// required pairs ` `int` `countPairs(``int` `n) ` `{ ` ` `  `    ``// Special case ` `    ``if` `(n == 2) ` `        ``return` `4; ` ` `  `    ``// Number which will give the max value ` `    ``// for ((n % i) % j) % n ` `    ``int` `num = ((n / 2) + 1); ` ` `  `    ``// To store the maximum possible value  ` `    ``// of ((n % i) % j) % n ` `    ``int` `max = n % num; ` ` `  `    ``// Count of possible pairs ` `    ``int` `count = n - max; ` ` `  `    ``return` `count; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `n = 5; ` `    ``cout << countPairs(n); ` `} ` ` `  `// This code is contributed by Code_Mech. `

## Java

 `// Java implementation of the approach ` `class` `GFG { ` ` `  `    ``// Function to return the count of required pairs ` `    ``public` `static` `int` `countPairs(``int` `n) ` `    ``{ ` ` `  `        ``// Special case ` `        ``if` `(n == ``2``) ` `            ``return` `4``; ` ` `  `        ``// Number which will give the max value ` `        ``// for ((n % i) % j) % n ` `        ``int` `num = ((n / ``2``) + ``1``); ` ` `  `        ``// To store the maximum possible value of ` `        ``// ((n % i) % j) % n ` `        ``int` `max = n % num; ` ` `  `        ``// Count of possible pairs ` `        ``int` `count = n - max; ` ` `  `        ``return` `count; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `n = ``5``; ` `        ``System.out.println(countPairs(n)); ` `    ``} ` `} `

## Python3

 `# Python3 implementation of the approach ` ` `  `# Function to return the count of required pairs ` `def` `countPairs(n): ` `     `  `    ``# Special case ` `    ``if` `(n ``=``=` `2``): ` `        ``return` `4` ` `  `    ``# Number which will give the max value ` `    ``# for ((n % i) % j) % n ` `    ``num ``=` `((n ``/``/` `2``) ``+` `1``); ` ` `  `    ``# To store the maximum possible value  ` `    ``# of ((n % i) % j) % n ` `    ``max` `=` `n ``%` `num; ` ` `  `    ``# Count of possible pairs ` `    ``count ``=` `n ``-` `max``; ` ` `  `    ``return` `count ` ` `  `# Driver code ` `if` `__name__ ``=``=``"__main__"` `: ` ` `  `    ``n ``=` `5``; ` `print``(countPairs(n)); ` ` `  `# This code is contributed by Code_Mech `

## C#

 `// C# implementation of above approach ` `using` `System; ` ` `  `class` `GFG ` `{ ` `     `  `    ``// Function to return the count of required pairs ` `    ``static` `int` `countPairs(``int` `n) ` `    ``{ ` ` `  `        ``// Special case ` `        ``if` `(n == 2) ` `            ``return` `4; ` ` `  `        ``// Number which will give the max value ` `        ``// for ((n % i) % j) % n ` `        ``int` `num = ((n / 2) + 1); ` ` `  `        ``// To store the maximum possible value of ` `        ``// ((n % i) % j) % n ` `        ``int` `max = n % num; ` ` `  `        ``// Count of possible pairs ` `        ``int` `count = n - max; ` ` `  `        ``return` `count; ` `    ``} ` ` `  `    ``// Driver code ` `    ``static` `public` `void` `Main () ` `    ``{ ` `            ``int` `n = 5; ` `        ``Console.WriteLine(countPairs(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by Tushil..  `

## PHP

 ` `

Output:

```3
```

Time Complexity: O(1)

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