Given a number n, we need to find the number of ordered pairs of a and b such gcd(a, b) is b itself
Examples :
Input : n = 2
Output : 3
The pairs are (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] Counting GCD Pairs by Divisor Property
gcd(a, b) = b means b is a factor of a. So the 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.
[Expected Approach] Counting GCD Pairs Using Divisor Multiples and Optimized Summation
gcd(a, b) = b means that a is a multiple of b. So the 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, a 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;
}
// Java implementation of counting pairs
// such that gcd (a, b) = b
import java.io.*;
public class GFG {
// returns number of valid pairs
static 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 code
public static void main(String[] args) {
System.out.println(CountPairs(1));
System.out.println(CountPairs(2));
System.out.println(CountPairs(3));
}
}
// This code is contributed by Anant Agarwal.
# Python implementation of counting
# pairs such that gcd (a, b) = b
# returns number of valid pairs
def CountPairs(n):
# initialize k
k = n
# loop till imin <= n
imin = 1
# Initialize result
ans = 0
while(imin <= n):
# max i with given k floor(n / k)
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 code
print(CountPairs(1))
print(CountPairs(2))
print(CountPairs(3))
# This code is contributed by Anant Agarwal.
// C# implementation of counting
// pairs such that gcd (a, b) = b
using System;
class GFG {
// returns number of valid pairs
static 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 code
public static void Main(String []args)
{
Console.WriteLine(CountPairs(1));
Console.WriteLine(CountPairs(2));
Console.WriteLine(CountPairs(3));
}
}
// This code is contributed by vt_m.
<script>
// Javascript implementation of counting pairs
// such that gcd (a, b) = b
// returns number of valid pairs
function CountPairs(n)
{
// initialize k
let k = n;
// loop till imin <= n
let imin = 1;
// Initialize result
let ans = 0;
while (imin <= n) {
// max i with given k floor(n/k)
let imax = Math.floor(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 = Math.floor(n / imin);
}
return ans;
}
// Driver function
document.write(CountPairs(1) + "<br>");
document.write(CountPairs(2) + "<br>");
document.write(CountPairs(3) + "<br>");
// This is code is contributed by Mayank Tyagi
</script>
<?php
// PHP implementation of counting
// pairs such that gcd (a, b) = b
// returns number of valid pairs
function CountPairs($n)
{
// initialize k
$k = $n;
// loop till imin <= n
$imin = 1;
// Initialize result
$ans = 0;
while ($imin <= $n)
{
// max i with given k floor(n/k)
$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;
$k = (int)($n / $imin);
}
return $ans;
}
// Driver Code
echo(CountPairs(1) . "\n");
echo(CountPairs(2) . "\n");
echo(CountPairs(3) . "\n");
// This code is contributed by Ajit.
?>
Output
1 3 5
Time complexity: O(n). This is because the while loop takes O(n) time to complete since it is looping over all elements of the array.
Auxiliary space: O(1), as no extra space is used.