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# Count of pairs from first N natural numbers with remainder at least K

• Difficulty Level : Hard
• Last Updated : 06 Aug, 2021

Given two positive integers N and K, the task is to find the number of pairs (a, b) over the range [1, N] such that a%b is at least K.

Examples:

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Input: N = 5, K = 2
Output: 7
Explanation:
Following are the all possible pairs satisfying the given criteria:

1. (2, 3): The value of 2%3 = 2(>= K).
2. (5, 3): The value of 5%3 = 2(>= K).
3. (2, 4): The value of 2%4 = 2(>= K).
4. (3, 4): The value of 3%4 = 3(>= K).
5. (2, 5): The value of 2%5 = 2(>= K).
6. (3, 5): The value of 3%5 = 3(>= K).
7. (4, 5): The value of 4%5 = 4(>= K).

Therefore, the total count of pairs is 7.

Input: N = 6, K = 0
Output: 36

Naive Approach: The simplest approach to solve the given problem is to generate all possible pairs (a, b) over the range [1, N] and if the value of a%b is at least K, then count this pair. After checking for all the pairs, print the total pairs obtained.

Time Complexity: O(N2)
Auxiliary Space: O(1)

Efficient Approach: The above approach can also be optimized by iterating over the range [1, N] and fix the second number in the pair, i.e., b. For each fixed b there will be a period of N/b and every period can be combiner with (b – K) elements. So, a total of (N/b)*(b – K) elements will be there. Now for the remaining elements that are N%b there will be max(0, n%b – k + 1) pairs. Follow the steps below to solve the problem:

• If the value of K is 0, then print N2 as the resultant number of valid pairs.
• Initialize the variable, say ans as 0 that stores the resultant count of pairs.
• Iterate over the range [K + 1, N] using the variable b and perform the following steps:
• Add the value of (N/b)*(b – K) to the variable ans.
• Add the value of the maximum of (N % b – K + 1) or 0 to the variable ans.
• After performing the above steps, print the value of ans as the resultant count of pairs.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach` `#include ``using` `namespace` `std;` `// Function to count the number of pairs``// (a, b) such that a%b is at least K``int` `countTotalPairs(``int` `N, ``int` `K)``{``    ``// Base Case``    ``if` `(K == 0) {``        ``return` `N * N;``    ``}` `    ``// Stores resultant count of pairs``    ``int` `ans = 0;` `    ``// Iterate over the range [K + 1, N]``    ``for` `(``int` `b = K + 1; b <= N; b++) {` `        ``// Find the cycled elements``        ``ans += (N / b) * (b - K);` `        ``// Find the remaining elements``        ``ans += max(N % b - K + 1, 0);``    ``}` `    ``// Return the resultant possible``    ``// count of pairs``    ``return` `ans;``}` `// Driver Code``int` `main()``{``    ``int` `N = 5, K = 2;``    ``cout << countTotalPairs(N, K);` `    ``return` `0;``}`

## Java

 `/*package whatever //do not write package name here */``import` `java.io.*;` `class` `GFG {` `  ``// Function to count the number of pairs``  ``// (a, b) such that a%b is at least K``  ``public` `static` `int` `countTotalPairs(``int` `N, ``int` `K)``  ``{` `    ``// Base case``    ``if` `(K == ``0``) {``      ``return` `N * N;``    ``}` `    ``// Stores resultant count of pairs``    ``int` `ans = ``0``;` `    ``// Iterate over the range [K + 1, N]``    ``for` `(``int` `i = K + ``1``; i <= N; i++)``    ``{` `      ``// Find the cycled element``      ``ans += (N / i) * (i - K);``      ``if` `((N % i) - K + ``1` `> ``0``)``      ``{` `        ``// Find the remaining element``        ``ans += (N % i) - K + ``1``;``      ``}``    ``}` `    ``// Return the resultant possible``    ``// count of pairs``    ``return` `ans;``  ``}` `  ``// Driver code``  ``public` `static` `void` `main(String[] args)``  ``{``    ``int` `N = ``5``, K = ``2``;``    ``System.out.println(countTotalPairs(N, K));``  ``}``}` `// This code is contributed by maddler.`

## Python3

 `# Python program for the above approach;` `# Function to count the number of pairs``# (a, b) such that a%b is at least K``def` `countTotalPairs(N, K):``    ``# Base Case``    ``if` `(K ``=``=` `0``) :``        ``return` `N ``*` `N``    `  `    ``# Stores resultant count of pairs``    ``ans ``=` `0` `    ``# Iterate over the range [K + 1, N]``    ``for` `b ``in` `range``(K ``+` `1``, N ``+` `1``) :` `        ``# Find the cycled elements``        ``ans ``+``=` `(N ``/``/` `b) ``*` `(b ``-` `K)` `        ``# Find the remaining elements``        ``ans ``+``=` `max``(N ``%` `b ``-` `K ``+` `1``, ``0``)``    `  `    ``# Return the resultant possible``    ``# count of pairs``    ``return` `ans`  `# Driver Code` `N ``=` `5``K ``=` `2``print``(countTotalPairs(N, K))` `# This code is contributed by _saurabh_jaiswal`

## C#

 `// C# program for the above approach``using` `System;``using` `System.Collections.Generic;` `class` `GFG{` `// Function to count the number of pairs``// (a, b) such that a%b is at least K``static` `int` `countTotalPairs(``int` `N, ``int` `K)``{``    ``// Base Case``    ``if` `(K == 0) {``        ``return` `N * N;``    ``}` `    ``// Stores resultant count of pairs``    ``int` `ans = 0;` `    ``// Iterate over the range [K + 1, N]``    ``for` `(``int` `b = K + 1; b <= N; b++) {` `        ``// Find the cycled elements``        ``ans += (N / b) * (b - K);` `        ``// Find the remaining elements``        ``ans += Math.Max(N % b - K + 1, 0);``    ``}` `    ``// Return the resultant possible``    ``// count of pairs``    ``return` `ans;``}` `// Driver Code``public` `static` `void` `Main()``{``    ``int` `N = 5, K = 2;``    ``Console.Write(countTotalPairs(N, K));``}``}` `// This code is contributed by ipg2016107.`

## Javascript

 ``
Output:
`7`

Time Complexity: O(N)
Auxiliary Space: O(1)

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