Count number of pairs (A <= N, B <= N) such that gcd (A , B) is B

2

Given a number n. We need find the number of ordered pairs of a and b such gcd(a, b) is b itself

Examples:

Input : n = 2
Output : 3
(1, 1) (2, 2) and (2, 1) 

Input : n = 3
Output : 5
(1, 1) (2, 2) (3, 3) (2, 1) and (3, 1)

Naive approach : gcd(a, b) = b means b is a factor of a. So total number of pairs will be equal to sum of divisors for each a = 1 to n. Please refer find all divisors of a natural number for implementation.

Efficient approach : gcd(a, b) = b means that a is a multiple of b. So total number of pairs will be sum of number of multiples of each b (where b varies from 1 to n) which are less than or equal to n.
For a number i, number of multiples of i is less than or equal to floor(n/i). So what we need to do is just sum the floor(n/i) for each i = 1 to n and print it. But more optimizations can be done. floor(n/i) can have atmost 2*sqrt(n) values for i >= sqrt(n). floor(n/i) can vary from 1 to sqrt(n) and similarly for i = 1 to sqrt(n) floor(n/i) can have values from 1 to sqrt(n). So total of 2*sqrt(n) distinct values

let floor(n/i) = k
k <= n/i < k + 1
n/k+1 < i <= n/k
floor(n/k+1) < i <= floor(n/k)
Thus for given k the largest value of i for 
which the floor(n/i) = k is floor(n/k)
and all the set of i for which the 
floor(n/i) = k are consecutive
// C++ implementation of counting pairs
// such that gcd (a, b) = b
#include <bits/stdc++.h>
using namespace std;

// returns number of valid pairs
int CountPairs(int n)
{
    // initialize k
    int k = n;

    // loop till imin <= n
    int imin = 1;

    // Initialize result
    int ans = 0;

    while (imin <= n) {

        // max i with given k floor(n/k)
        int imax = n / k;

        // adding k*(number of i with
        // floor(n/i) = k to ans
        ans += k * (imax - imin + 1);

        // set imin = imax + 1 and k = n/imin
        imin = imax + 1;
        k = n / imin;
    }

    return ans;
}

// Driver function
int main()
{
    cout << CountPairs(1) << endl;
    cout << CountPairs(2) << endl;
    cout << CountPairs(3) << endl;
    return 0;
}

Output:

1
3
5

This article is contributed by Ayush Jha. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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