Find the Nth natural number which is not divisible by A

Given two integers A and N, our task is to find the Nth natural number which is not divisible by A.

Examples:

Input: A = 4, N = 12 
Output: 15 
Explanation:
The series starting from 1 excluding the multiples of A would be 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17 and the 12th term which is not divisible by 4 is 15.

Input: A = 3, N = 20 
Output: 29 
Explanation:
The series starting from 1 excluding the multiples of A would be 1, 2, 4, 5, 7, 8, 10, 11 and so on and the Nth number which is not divisible by 3 is 29. 

Approach:

To solve the problem mentioned above we have to observe that there is a gap after every (A – 1) integer as the multiples of A are skipped. To find that the number N lies in which set we divide N by (A – 1), and store it in a variable lets say quotient. Now multiplying A with that variable and adding the remainder to it we will get the resultant answer. 

If A = 4, N = 7:
quotient = 7 / 3 = 2 
remainder = 7 % 3 = 1

So the answer is 
(A * quotient) + remainder 
= 4 * 2 + 1 = 9

But if the remainder is 0 it means it is the last element in the set. Lets us understand it by an example. 

If A = 4, N = 6:
quotient = 6 / 3 = 2 
remainder = 6 % 3 = 0

So the answer is 
(A * quotient) - 1 
= 4 * 2 - 1 = 7

Below is the implementation of the above approach:

C++

// C++ code to Find the Nth number which
// is not divisible by A from the series
#include <bits/stdc++.h>
using namespace std;

void findNum(int n, int k)
{

    // Find the quotient and the remainder
    // when k is divided by n-1
    int q = k / (n - 1);
    int r = k % (n - 1);
    int a;

    // If the remainder is not 0
    // multiply n by q and subtract 1
    // if remainder is 0 then
    // multiply n by q
    // and add the remainder
    if (r != 0)
        a = (n * q) + r;
    else
        a = (n * q) - 1;

    // Print the answer
    cout << a;
}

// Driver code
int main()
{
    int A = 4, N = 6;

    findNum(A, N);
    return 0;
}

// This code is contributed by PratikBasu

Java

// Java code to Find the Nth number which
// is not divisible by A from the series
class GFG {

    static void findNum(int n, int k)
    {

        // Find the quotient and the remainder
        // when k is divided by n-1
        int q = k / (n - 1);
        int r = k % (n - 1);
        int a = 0;

        // If the remainder is not 0
        // multiply n by q and subtract 1
        // if remainder is 0 then
        // multiply n by q
        // and add the remainder
        if (r != 0)
            a = (n * q) + r;
        else
            a = (n * q) - 1;

        // Print the answer
        System.out.println(a);
    }

    // Driver code
    public static void main(String[] args)
    {
        int A = 4;
        int N = 6;

        findNum(A, N);
    }
}

// This code is contributed by 29AjayKumar

Python3

# Python3 code to Find the Nth number which
# is not divisible by A from the series


def findNum(n, k):

    # Find the quotient and the remainder
    # when k is divided by n-1
    q = k//(n-1)
    r = k % (n-1)

    # if the remainder is not 0
    # multiply n by q and subtract 1

    # if remainder is 0 then
    # multiply n by q
    # and add the remainder
    if(r != 0):
        a = (n * q)+r
    else:
        a = (n * q)-1

    # print the answer
    print(a)


# driver code
A = 4
N = 6
findNum(A, N)

C#

// C# code to find the Nth number which
// is not divisible by A from the series
using System;

class GFG {

    static void findNum(int n, int k)
    {

        // Find the quotient and the remainder
        // when k is divided by n-1
        int q = k / (n - 1);
        int r = k % (n - 1);
        int a = 0;

        // If the remainder is not 0
        // multiply n by q and subtract 1
        // if remainder is 0 then
        // multiply n by q
        // and add the remainder
        if (r != 0)
            a = (n * q) + r;
        else
            a = (n * q) - 1;

        // Print the answer
        Console.WriteLine(a);
    }

    // Driver code
    public static void Main(String[] args)
    {
        int A = 4;
        int N = 6;

        findNum(A, N);
    }
}

// This code is contributed by amal kumar choubey

Javascript

<script>

// Javascript code to Find the Nth number which
// is not divisible by A from the series
function findNum(n, k)
{

    // Find the quotient and the remainder
    // when k is divided by n-1
    let q = parseInt(k / (n - 1));
    let r = k % (n - 1);
    let a;

    // If the remainder is not 0
    // multiply n by q and subtract 1
    // if remainder is 0 then
    // multiply n by q
    // and add the remainder
    if (r != 0)
        a = (n * q) + r;
    else
        a = (n * q) - 1;

    // Print the answer
    document.write(a);
}

// Driver code
let A = 4, N = 6;

findNum(A, N);

// This code is contributed by rishavmahato348

</script>

Output:

7

Time complexity: O(1)
Auxiliary space: O(1)