What is Count total divisors of A or B in a given range?

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Count total divisors of A or B in a given range️‍🔥

Given four integers m, n, a, b. Find how many integers from range m to n are divisible by a or b.
Examples :

Input: 3 11 2 3
Output: 6
Explanation:
m = 3, n = 11, a = 2, b = 3
There are total 6 numbers from 3 to 
11 which are divisible by 2 or 3 i.e, 
3, 4, 6, 8, 9, 10 

Input: arr[] = {11, 1000000, 6, 35}
Output: 190475

A Naive approach is to run a loop from m to n and count all numbers which are divisible by either a or b. Time complexity of this approach will be O(m – n) which will definitely time out for large values of m.
An efficient approach is to use simple LCM and division method.

Divide n by a to obtain total count of all numbers(1 to n) divisible by ‘a’.
Divide m-1 by a to obtain total count of all numbers(1 to m-1) divisible by ‘a’.
Subtract the count of step 1 and 2 to obtain total divisors in range m to n.

Now we have a total number of divisors of ‘a’ in given range. Repeat the above to count total divisors of ‘b’.
Add these to obtain total count of divisors ‘a’ and ‘b’.
But the number divisible by both a and b counted twice. Therefore, to remove this ambiguity we can use LCM of a and b to count total number divisible by both ‘a’ and ‘b’.

Find LCM of ‘a’ and ‘b’.
Divide n by LCM to obtain the count of numbers(1 to n) divisible by both ‘a’ and ‘b’.
Divide m-1 by LCM to obtain the count of numbers(1 to m-1) divisible by both ‘a’ and ‘b’.
Subtract the count of steps 2 and 3 to obtain total divisors of both ‘a’ and ‘b’.

Now subtract this result from the previous calculated result to find total count of all unique divisors of ‘a’ or ‘b’.

C++

// C++ program to count total divisors of 'a'
// or 'b' in a given range 
#include <bits/stdc++.h>
using namespace std;
 
// Utility function to find LCM of two numbers
int FindLCM(int a, int b)
{
    return (a * b) / __gcd(a, b);
}
 
// Function to calculate all divisors in given range
int rangeDivisor(int m, int n, int a, int b)
{
    // Find LCM of a and b
    int lcm = FindLCM(a, b);
 
    int a_divisor = n / a - (m - 1) / a;
    int b_divisor = n / b - (m - 1) / b;
 
    // Find common divisor by using LCM
    int common_divisor = n / lcm - (m - 1) / lcm;
 
    int ans = a_divisor + b_divisor - common_divisor;
    return ans;
}
 
// Driver code
int main()
{
    int m = 3, n = 11, a = 2, b = 3;
    cout << rangeDivisor(m, n, a, b) << endl;
 
    m = 11, n = 1000000, a = 6, b = 35;
    cout << rangeDivisor(m, n, a, b);
    return 0;
}

Java

// Java program to count total divisors of 'a'
// or 'b' in a given range
 
import java.math.BigInteger;
 
class Test
{
    // Utility method to find LCM of two numbers
    static int FindLCM(int a, int b)
    {
        return (a * b) / new BigInteger(a+"").gcd(new BigInteger(b+"")).intValue();
    }
     
    // method to calculate all divisors in given range
    static int rangeDivisor(int m, int n, int a, int b)
    {
        // Find LCM of a and b
        int lcm = FindLCM(a, b);
      
        int a_divisor = n / a - (m - 1) / a;
        int b_divisor = n / b - (m - 1) / b;
      
        // Find common divisor by using LCM
        int common_divisor = n / lcm - (m - 1) / lcm;
      
        int ans = a_divisor + b_divisor - common_divisor;
        return ans;
    }
     
    // Driver method
    public static void main(String args[])
    {
        int m = 3, n = 11, a = 2, b = 3;
        System.out.println(rangeDivisor(m, n, a, b));
      
        m = 11; n = 1000000 ; a = 6; b = 35;
        System.out.println(rangeDivisor(m, n, a, b));
    }
}

Python3

# python program to count total divisors
# of 'a' or 'b' in a given range 
 
def __gcd(x, y):
 
    if x > y:
        small = y
    else:
        small = x
    for i in range(1, small+1):
        if((x % i == 0) and (y % i == 0)):
            gcd = i
             
    return gcd
     
# Utility function to find LCM of two
# numbers
def FindLCM(a, b):
    return (a * b) / __gcd(a, b);
 
 
# Function to calculate all divisors in
# given range
def rangeDivisor(m, n, a, b):
     
    # Find LCM of a and b
    lcm = FindLCM(a, b)
 
    a_divisor = int( n / a - (m - 1) / a)
    b_divisor = int(n / b - (m - 1) / b)
     
    # Find common divisor by using LCM
    common_divisor =int( n / lcm - (m - 1) / lcm)
 
    ans = a_divisor + b_divisor - common_divisor
    return ans
 
# Driver code
m = 3
n = 11
a = 2
b = 3;
print(rangeDivisor(m, n, a, b))
m = 11
n = 1000000
a = 6
b = 35
print(rangeDivisor(m, n, a, b))
 
# This code is contributed by Sam007

C#

// C# program to count total divisors
// of 'a' or 'b' in a given range 
using System;
 
class GFG {
 
    static int GCD(int num1, int num2)
    {
        int Remainder;
 
        while (num2 != 0)
        {
            Remainder = num1 % num2;
            num1 = num2;
            num2 = Remainder;
        }
 
        return num1;
    }
     
    // Utility function to find LCM of
    // two numbers
    static int FindLCM(int a, int b)
    {
        return (a * b) / GCD(a, b);
    }
     
    // Function to calculate all divisors in given range
    static int rangeDivisor(int m, int n, int a, int b)
    {
        // Find LCM of a and b
        int lcm = FindLCM(a, b);
     
        int a_divisor = n / a - (m - 1) / a;
        int b_divisor = n / b - (m - 1) / b;
     
        // Find common divisor by using LCM
        int common_divisor = n / lcm - (m - 1) / lcm;
     
        int ans = a_divisor + b_divisor - common_divisor;
        return ans;
    }
         
    public static void Main ()
    {
        int m = 3, n = 11, a = 2, b = 3;
        Console.WriteLine(rangeDivisor(m, n, a, b));
 
        m = 11;    n = 1000000;
        a = 6; b = 35;
        Console.WriteLine(rangeDivisor(m, n, a, b));
    }
}
 
// This code is contributed by Sam007.

PHP

<?php
// PHP program to count total
// divisors of 'a' or 'b' in 
// a given range 
 
function gcd($a, $b) 
{
    return ($a % $b) ? 
    gcd($b, $a % $b) : 
                   $b;
}
 
 
// Utility function to 
// find LCM of two numbers
function FindLCM($a, $b)
{
    return ($a * $b) / gcd($a, $b);
}
 
// Function to calculate 
// all divisors in given range
function rangeDivisor($m, $n, $a, $b)
{
    // Find LCM of a and b
    $lcm = FindLCM($a, $b);
 
    $a_divisor = $n / $a - ($m - 1) / $a;
    $b_divisor = $n / $b - ($m - 1) / $b;
 
    // Find common divisor by using LCM
    $common_divisor = $n / $lcm - 
                          ($m - 1) / $lcm;
 
    $ans = $a_divisor + $b_divisor - 
                        $common_divisor;
    return $ans;
}
 
// Driver Code
$m = 3;
$n = 11;
$a = 2;
$b = 3;
print(ceil(rangeDivisor($m, $n, $a, $b)));
echo "\n";
 
$m = 11;
$n = 1000000;
$a = 6;
$b = 35;
print(ceil(rangeDivisor($m, $n,$a,$b)));
 
// This code is contributed by Sam007
?>

Javascript

<script>
 
    // JavaScript program to count total divisors
    // of 'a' or 'b' in a given range 
     
    function GCD(num1, num2)
    {
        let Remainder;
   
        while (num2 != 0)
        {
            Remainder = num1 % num2;
            num1 = num2;
            num2 = Remainder;
        }
   
        return num1;
    }
       
    // Utility function to find LCM of
    // two numbers
    function FindLCM(a, b)
    {
        return parseInt((a * b) / GCD(a, b), 10);
    }
       
    // Function to calculate all divisors in given range
    function rangeDivisor(m, n, a, b)
    {
        // Find LCM of a and b
        let lcm = FindLCM(a, b);
       
        let a_divisor = parseInt(n / a, 10) - 
                        parseInt((m - 1) / a, 10);
                         
        let b_divisor = parseInt(n / b, 10) - 
                        parseInt((m - 1) / b, 10);
       
        // Find common divisor by using LCM
        let common_divisor = parseInt(n / lcm, 10) - 
                             parseInt((m - 1) / lcm, 10);
       
        let ans = a_divisor + b_divisor - common_divisor;
        return ans;
    }
     
    let m = 3, n = 11, a = 2, b = 3;
    document.write(rangeDivisor(m, n, a, b) + "</br>");
 
    m = 11;    n = 1000000;
    a = 6; b = 35;
    document.write(rangeDivisor(m, n, a, b));
     
</script>

Output:

6
190475

Time complexity: O(log(MAX(a, b))
Auxiliary space: O(1)