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.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.