Find and Count total factors of co-prime A or B in a given range 1 to N

Given three integers N, A, B, the task is to find the remainder when the sum of integers which are divisible by either A or B in the range [1, N] is divided by the number of integers in this range.

Note: The numbers A and B are co-primes.

Examples:

Input: N = 88, A = 11, B = 8
Output: 8
Explanation:
There are a total of 18 numbers in the range [1, 88] which are divisible by either 8 or 11. They are:
{ 8, 11, 16, 22, 24, 32, 33, 40, 44, 48, 55, 56, 64, 66, 72, 77, 80, 88 }. Therefore, the sum of these numbers is 836. Therefore, 836 % 18 = 8.

Input: N = 100, A = 7, B = 19
Output: 13
Explanation:
There are a total of 19 numbers in the range [1, 100] which are divisible by either 7 or 19. They are:
{ 7, 14, 19, 21, 28, 35, 38, 42, 49, 56, 57, 63, 70, 76, 77, 84, 91, 95, 98 }. Therefore, the sum of these numbers is 1020. Therefore, 1020 % 19 = 13.



Naive Approach: The naive approach is to run a loop from 1 to N and count all the numbers which are divisible by either A or B while simultaneously adding those numbers in a variable to find its sum.

Time Complexity: O(N)

Efficient Approach: Efficient approach is to use the division method.

  • By using the division method, the count of the numbers which are divisible either by A or B can be found in the constant time. The idea is to:
    1. Divide N by A to get the count of numbers divisible by A in the range [1, N].
    2. Divide N by B to get the count of numbers divisible by B in the range [1, N].
    3. Divide N by A * B to get the count of numbers divisible by both A and B.
    4. Add the values obtained in step 1 and step 2 and subtract the value obtained in step 3 to remove the numbers which have been counted twice.
  • Since we are even interested in finding the numbers which are divisible in this range, the idea is to reduce the number of times the conditions are checked by the following way:
    • Instead of completely relying on one loop, we can use two loops.
    • One loop is to find the numbers divisible by A. Instead of incrementing the values by 1, we start the loop from A and increment it by A. This reduces the number of comparisons drastically.
    • Similarly, another loop is used to find the numbers divisible by B.
    • Since again, there might be repetitions in the numbers, the numbers are stored in a set so that there are no repetitions.
  • Once the count and sum of the numbers are found, then directly modulo operation can be applied to compute the final answer.

Below is the implementation of the above approach:

CPP

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// C++ implementation of the above approach
#include <algorithm>
#include <iostream>
#include <set>
#define ll long long
using namespace std;
  
// Function to return the count of numbers
// which are divisible by both A and B in
// the range [1, N] in constant time
ll int countOfNum(ll int n, ll int a, ll int b)
{
    ll int cnt_of_a, cnt_of_b, cnt_of_ab, sum;
  
    // Compute the count of numbers divisible by
    // A in the range [1, N]
    cnt_of_a = n / a;
  
    // Compute the count of numbers divisible by
    // B in the range [1, N]
    cnt_of_b = n / b;
  
    // Adding the counts which are
    // divisible by A and B
    sum = cnt_of_b + cnt_of_a;
  
    // The above value might contain repeated
    // values which are divisible by both
    // A and B. Therefore, the count of numbers
    // which are divisible by both A and B are found
    cnt_of_ab = n / (a * b);
  
    // The count computed above is subtracted to
    // compute the final count
    sum = sum - cnt_of_ab;
  
    return sum;
}
  
// Function to return the sum of numbers
// which are divisible by both A and B
// in the range [1, N]
ll int sumOfNum(ll int n, ll int a, ll int b)
{
    ll int i;
    ll int sum = 0;
  
    // Set to store the numbers so that the
    // numbers are not repeated
    set<ll int> ans;
  
    // For loop to find the numbers
    // which are divisible by A and insert
    // them into the set
    for (i = a; i <= n; i = i + a) {
        ans.insert(i);
    }
  
    // For loop to find the numbers
    // which are divisible by A and insert
    // them into the set
    for (i = b; i <= n; i = i + b) {
        ans.insert(i);
    }
  
    // For loop to iterate through the set
    // and find the sum
    for (auto it = ans.begin();
         it != ans.end(); it++) {
        sum = sum + *it;
    }
  
    return sum;
}
  
// Driver code
int main()
{
    ll int N = 88;
    ll int A = 11;
    ll int B = 8;
  
    ll int count = countOfNum(N, A, B);
    ll int sumofnum = sumOfNum(N, A, B);
  
    cout << sumofnum % count << endl;
  
    return 0;
}

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Java

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// Java implementation of the above approach
  
import java.util.*;
// Function to return the count of numbers
// which are divisible by both A and B in
// the range [1, N] in constant time
  
class GFG 
{
  
    static int countOfNum( int n,  int a, int b)
    {
        int cnt_of_a, cnt_of_b, cnt_of_ab, sum;
      
        // Compute the count of numbers divisible by
        // A in the range [1, N]
        cnt_of_a = n / a;
      
        // Compute the count of numbers divisible by
        // B in the range [1, N]
        cnt_of_b = n / b;
      
        // Adding the counts which are
        // divisible by A and B
        sum = cnt_of_b + cnt_of_a;
      
        // The above value might contain repeated
        // values which are divisible by both
        // A and B. Therefore, the count of numbers
        // which are divisible by both A and B are found
        cnt_of_ab = n / (a * b);
      
        // The count computed above is subtracted to
        // compute the final count
        sum = sum - cnt_of_ab;
      
        return sum;
    }
      
    // Function to return the sum of numbers
    // which are divisible by both A and B
    // in the range [1, N]
    static int sumOfNum( int n,  int a, int b)
    {
        int i;
        int sum = 0;
      
        // Set to store the numbers so that the
        // numbers are not repeated
        Set< Integer> ans =  new HashSet<Integer>();
      
        // For loop to find the numbers
        // which are divisible by A and insert
        // them into the set
        for (i = a; i <= n; i = i + a) {
            ans.add(i);
        }
      
        // For loop to find the numbers
        // which are divisible by A and insert
        // them into the set
        for (i = b; i <= n; i = i + b) {
            ans.add(i);
        }
      
        // For loop to iterate through the set
        // and find the sum
        for (Integer it : ans) {
            sum = sum + it;
        }
      
        return sum;
    }
      
    // Driver code
    public static void main (String []args)
    {
        int N = 88;
        int A = 11;
        int B = 8;
      
        int count = countOfNum(N, A, B);
        int sumofnum = sumOfNum(N, A, B);
      
        System.out.print(sumofnum % count);
    
}
  
// This code is contributed by chitranayal  

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Python3

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# Python3 implementation of the above approach
  
# Function to return the count of numbers
# which are divisible by both A and B in
# the range [1, N] in constant time
def countOfNum(n, a, b):
    cnt_of_a, cnt_of_b, cnt_of_ab, sum = 0, 0, 0, 0
  
    # Compute the count of numbers divisible by
    # A in the range [1, N]
    cnt_of_a = n // a
  
    # Compute the count of numbers divisible by
    # B in the range [1, N]
    cnt_of_b = n // b
  
    # Adding the counts which are
    # divisible by A and B
    sum = cnt_of_b + cnt_of_a
  
    # The above value might contain repeated
    # values which are divisible by both
    # A and B. Therefore, the count of numbers
    # which are divisible by both A and B are found
    cnt_of_ab = n // (a * b)
  
    # The count computed above is subtracted to
    # compute the final count
    sum = sum - cnt_of_ab
  
    return sum
  
# Function to return the sum of numbers
# which are divisible by both A and B
# in the range [1, N]
def sumOfNum(n, a, b):
  
    i = 0
    sum = 0
  
    # Set to store the numbers so that the
    # numbers are not repeated
    ans = dict()
  
    # For loop to find the numbers
    # which are divisible by A and insert
    # them into the set
    for i in range(a, n + 1, a):
        ans[i] = 1
  
    # For loop to find the numbers
    # which are divisible by A and insert
    # them into the set
    for i in range(b, n + 1, b):
        ans[i] = 1
  
    # For loop to iterate through the set
    # and find the sum
    for it in ans:
        sum = sum + it
    return sum
  
# Driver code
if __name__ == '__main__':
    N = 88
    A = 11
    B = 8
  
    count = countOfNum(N, A, B)
    sumofnum = sumOfNum(N, A, B)
  
    print(sumofnum % count)
  
# This code is contributed by mohit kumar 29

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C#

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// C# implementation of the above approach
using System;
using System.Collections.Generic;
  
class GFG 
{
    // Function to return the count of numbers
    // which are divisible by both A and B in
    // the range [1, N] in constant time
   
    static int countOfNum( int n,  int a, int b)
    {
        int cnt_of_a, cnt_of_b, cnt_of_ab, sum;
       
        // Compute the count of numbers divisible by
        // A in the range [1, N]
        cnt_of_a = n / a;
       
        // Compute the count of numbers divisible by
        // B in the range [1, N]
        cnt_of_b = n / b;
       
        // Adding the counts which are
        // divisible by A and B
        sum = cnt_of_b + cnt_of_a;
       
        // The above value might contain repeated
        // values which are divisible by both
        // A and B. Therefore, the count of numbers
        // which are divisible by both A and B are found
        cnt_of_ab = n / (a * b);
       
        // The count computed above is subtracted to
        // compute the readonly count
        sum = sum - cnt_of_ab;
       
        return sum;
    }
       
    // Function to return the sum of numbers
    // which are divisible by both A and B
    // in the range [1, N]
    static int sumOfNum( int n,  int a, int b)
    {
        int i;
        int sum = 0;
       
        // Set to store the numbers so that the
        // numbers are not repeated
        HashSet< int> ans =  new HashSet<int>();
       
        // For loop to find the numbers
        // which are divisible by A and insert
        // them into the set
        for (i = a; i <= n; i = i + a) {
            ans.Add(i);
        }
       
        // For loop to find the numbers
        // which are divisible by A and insert
        // them into the set
        for (i = b; i <= n; i = i + b) {
            ans.Add(i);
        }
       
        // For loop to iterate through the set
        // and find the sum
        foreach (int it in ans) {
            sum = sum + it;
        }
       
        return sum;
    }
       
    // Driver code
    public static void Main(String []args)
    {
        int N = 88;
        int A = 11;
        int B = 8;
       
        int count = countOfNum(N, A, B);
        int sumofnum = sumOfNum(N, A, B);
       
        Console.Write(sumofnum % count);
    
}
  
// This code is contributed by 29AjayKumar

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Output:

8

Time Complexity Analysis:

  • The time taken to run the for loop to find the numbers which are divisible by A is O(N / A).
  • The time taken to run the for loop to find the numbers which are divisible by B is O(N / B).
  • Therefore, overall time complexity is O(N / A) + O(N / B).

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