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Find the Nth term divisible by a or b or c

  • Difficulty Level : Medium
  • Last Updated : 10 May, 2021

Given four integers a, b, c, and N. The task is to find the Nth term which is divisible by either of a, b or c.
Examples: 
 

Input: a = 2, b = 3, c = 5, N = 10 
Output: 14 
Sequence is 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16… 
Input: a = 3, b = 5, c = 7, N = 10 
Output: 18 
 

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Naive Approach: A simple approach is to traverse over all the terms starting from 1 until we find the desired Nth term which is divisible by either a, b or c. This solution has time complexity of O(N).
Efficient Approach: The idea is to use Binary search. Here we can calculate how many numbers from 1 to num are divisible by either a, b or c by using the formula: (num / a) + (num / b) + (num / c) – (num / lcm(a, b)) – (num / lcm(b, c)) – (num / lcm(a, c)) + (num / lcm(a, b, c)) 
The above formula is derived using set theory
Below is the implementation of the above approach: 
 

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to return
// gcd of a and b
int gcd(int a, int b)
{
    if (a == 0)
        return b;
 
    return gcd(b % a, a);
}
 
// Function to return the lcm of a and b
int lcm(int a, int b)
{
    return (a * b) / gcd(a, b);
}
 
// Function to return the count of numbers
// from 1 to num which are divisible by a, b or c
int divTermCount(int a, int b, int c, int num)
{
 
    // Calculate number of terms divisible by a and
    // by b and by c then, remove the terms which is are
    // divisible by both a and b, both b and c, both
    // c and a and then add which are divisible by a and
    // b and c
    return ((num / a) + (num / b) + (num / c)
            - (num / lcm(a, b))
            - (num / lcm(b, c))
            - (num / lcm(a, c))
            + (num / lcm(a, lcm(b, c))));
}
 
// Function to find the nth term
// divisible by a, b or c
// by using binary search
int findNthTerm(int a, int b, int c, int n)
{
 
    // Set low to 1 and high to max (a, b, c) * n
    int low = 1, high = INT_MAX, mid;
 
    while (low < high) {
        mid = low + (high - low) / 2;
 
        // If the current term is less than
        // n then we need to increase low
        // to mid + 1
        if (divTermCount(a, b, c, mid) < n)
            low = mid + 1;
 
        // If current term is greater than equal to
        // n then high = mid
        else
            high = mid;
    }
 
    return low;
}
 
// Driver code
int main()
{
    int a = 2, b = 3, c = 5, n = 10;
 
    cout << findNthTerm(a, b, c, n);
 
    return 0;
}

Java




// Java implementation of the approach
class GFG
{
     
    // Function to return
    // gcd of a and b
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
         
        return gcd(b % a, a);
    }
 
    // Function to return the lcm of a and b
    static int lcm(int a, int b)
    {
        return (a * b) / gcd(a, b);
    }
 
    // Function to return the count of numbers
    // from 1 to num which are divisible by a, b or c
    static int divTermCount(int a, int b, int c, int num)
    {
 
        // Calculate number of terms divisible by a and
        // by b and by c then, remove the terms which is are
        // divisible by both a and b, both b and c, both
        // c and a and then add which are divisible by a and
        // b and c
        return ((num / a) + (num / b) + (num / c)
                - (num / lcm(a, b))
                - (num / lcm(b, c))
                - (num / lcm(a, c))
                + (num / lcm(a, lcm(b, c))));
    }
 
    // Function to find the nth term
    // divisible by a, b or c
    // by using binary search
    static int findNthTerm(int a, int b, int c, int n)
    {
 
        // Set low to 1 and high to max (a, b, c) * n
        int low = 1, high = Integer.MAX_VALUE, mid;
 
        while (low < high) {
            mid = low + (high - low) / 2;
 
            // If the current term is less than
            // n then we need to increase low
            // to mid + 1
            if (divTermCount(a, b, c, mid) < n)
                low = mid + 1;
     
            // If current term is greater than equal to
            // n then high = mid
            else
                high = mid;
        }
 
        return low;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        int a = 2, b = 3, c = 5, n = 10;
 
        System.out.println(findNthTerm(a, b, c, n));
 
    }
}
 
// This code is contributed by
// Rajnis09

Python




# Python3 implementation of the approach
 
# Function to return
# gcd of a and b
def gcd(a, b):
 
    if (a == 0):
        return b
 
    return gcd(b % a, a)
 
# Function to return the lcm of a and b
def lcm(a, b):
 
    return ((a * b) // gcd(a, b))
 
 
# Function to return the count of numbers
# from 1 to num which are divisible by a, b or c
def divTermCount(a, b, c, num):
 
    # Calculate number of terms divisible by a and
    # by b and by c then, remove the terms which is are
    # divisible by both a and b, both b and c, both
    # c and a and then add which are divisible by a and
    # b and c
    return ((num // a) + (num // b) + (num // c)
            - (num // lcm(a, b))
            - (num // lcm(b, c))
            - (num // lcm(a, c))
            + (num // lcm(a, lcm(b, c))))
 
 
# Function to find the nth term
# divisible by a, b or c
# by using binary search
def findNthTerm(a, b, c, n):
 
 
    # Set low to 1 and high to max (a, b, c) * n
    low = 1
    high = 10**9
    mid=0
 
    while (low < high):
        mid = low + (high - low) // 2
 
        # If the current term is less than
        # n then we need to increase low
        # to mid + 1
        if (divTermCount(a, b, c, mid) < n):
            low = mid + 1
 
        # If current term is greater than equal to
        # n then high = mid
        else:
            high = mid
 
 
    return low
 
# Driver code
a = 2
b = 3
c = 5
n = 10
 
print(findNthTerm(a, b, c, n))
 
# This code is contributed by mohit kumar 29

C#




// C# implementation of the approach
using System;
     
class GFG
{
     
    // Function to return
    // gcd of a and b
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
         
        return gcd(b % a, a);
    }
 
    // Function to return the lcm of a and b
    static int lcm(int a, int b)
    {
        return (a * b) / gcd(a, b);
    }
 
    // Function to return the count of numbers
    // from 1 to num which are divisible by a, b or c
    static int divTermCount(int a, int b, int c, int num)
    {
 
        // Calculate number of terms divisible by a and
        // by b and by c then, remove the terms which is are
        // divisible by both a and b, both b and c, both
        // c and a and then add which are divisible by a and
        // b and c
        return ((num / a) + (num / b) + (num / c)
                - (num / lcm(a, b))
                - (num / lcm(b, c))
                - (num / lcm(a, c))
                + (num / lcm(a, lcm(b, c))));
    }
 
    // Function to find the nth term
    // divisible by a, b or c
    // by using binary search
    static int findNthTerm(int a, int b, int c, int n)
    {
 
        // Set low to 1 and high to max (a, b, c) * n
        int low = 1, high = int.MaxValue, mid;
 
        while (low < high) {
            mid = low + (high - low) / 2;
 
            // If the current term is less than
            // n then we need to increase low
            // to mid + 1
            if (divTermCount(a, b, c, mid) < n)
                low = mid + 1;
     
            // If current term is greater than equal to
            // n then high = mid
            else
                high = mid;
        }
 
        return low;
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        int a = 2, b = 3, c = 5, n = 10;
 
        Console.WriteLine(findNthTerm(a, b, c, n));
 
    }
}
 
/* This code is contributed by PrinciRaj1992 */

Javascript




<script>
 
// Javascript implementation of the approach
 
// Function to return
// gcd of a and b
function gcd(a, b)
{
    if (a == 0)
        return b;
 
    return gcd(b % a, a);
}
 
// Function to return the lcm of a and b
function lcm(a, b)
{
    return parseInt((a * b) / gcd(a, b));
}
 
// Function to return the count of numbers
// from 1 to num which are divisible by a, b or c
function divTermCount(a, b, c, num)
{
 
    // Calculate number of terms divisible by a and
    // by b and by c then, remove the terms which is are
    // divisible by both a and b, both b and c, both
    // c and a and then add which are divisible by a and
    // b and c
    return (parseInt(num / a) + parseInt(num / b) +
    parseInt(num / c)
            - parseInt(num / lcm(a, b))
            - parseInt(num / lcm(b, c))
            - parseInt(num / lcm(a, c))
            + parseInt(num / lcm(a, lcm(b, c))));
}
 
// Function to find the nth term
// divisible by a, b or c
// by using binary search
function findNthTerm(a, b, c, n)
{
 
    // Set low to 1 and high to max (a, b, c) * n
    let low = 1, high = Number.MAX_VALUE, mid;
 
    while (low < high) {
        mid = low + parseInt((high - low) / 2);
 
        // If the current term is less than
        // n then we need to increase low
        // to mid + 1
        if (divTermCount(a, b, c, mid) < n)
            low = mid + 1;
 
        // If current term is greater than equal to
        // n then high = mid
        else
            high = mid;
    }
 
    return low;
}
 
// Driver code
    let a = 2, b = 3, c = 5, n = 10;
 
    document.write(findNthTerm(a, b, c, n));
 
</script>
Output: 
14

 




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