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Find Nth positive number whose digital root is X

  • Difficulty Level : Medium
  • Last Updated : 07 Jun, 2021

Given a number X ( 1<= X <= 9) and a positive number N, find the Nth positive number whose digital root is X.
Digital Root: The digital root of a positive number is obtained by iteratively summing up the digits of a number. On each iteration, the number is replaced by the sum of its digit and the iteration stops when the number is reduced to a single digit number. This single digit number is known as the digital root. For example, the digital root of 65 is 2, because 6 + 5 = 11 and 1 + 1 = 2.

Examples:  

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Input : X = 3, N = 100 
Output : 894 
The N-th Number whose digit root is X is 894



Input : X = 7, N = 43 
Output : 385

Simple Method: The simple method is to start from 1 and calculate the digital root of every number, whenever we come across a number whose digital root is equal to X, we increase our counter by 1. We will stop our search when our counter is equal to N.

Below is the implementation of the above approach:  

C++




// C++ program to find the N-th number whose
// digital root is X
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the digital root of
// a number
int findDigitalRoot(int num)
{
    int sum = INT_MAX, tempNum = num;
 
    while (sum >= 10) {
        sum = 0;
 
        while (tempNum > 0) {
            sum += tempNum % 10;
            tempNum /= 10;
        }
 
        tempNum = sum;
    }
 
    return sum;
}
 
// Function to find the Nth number with
// digital root as X
void findAnswer(int X, int N)
{
    // Counter variable to keep the
    // count of valid numbers
    int counter = 0;
 
    for (int i = 1; counter < N; ++i) {
 
        // Find digital root
        int digitalRoot = findDigitalRoot(i);
 
        // Check if is required answer or not
        if (digitalRoot == X) {
            ++counter;
        }
 
        // Print the answer if you have found it
        // and breakout of the loop
        if (counter == N) {
            cout << i;
 
            break;
        }
    }
}
 
// Driver Code
int main()
{
    int X = 1, N = 3;
 
    findAnswer(X, N);
 
    return 0;
}

Java




// Java program to find the N-th number whose
// digital root is X
class GFG
{
 
// Function to find the digital root of
// a number
static int findDigitalRoot(int num)
{
    int sum = Integer.MAX_VALUE, tempNum = num;
 
    while (sum >= 10)
    {
        sum = 0;
 
        while (tempNum > 0)
        {
            sum += tempNum % 10;
            tempNum /= 10;
        }
 
        tempNum = sum;
    }
 
    return sum;
}
 
// Function to find the Nth number with
// digital root as X
static void findAnswer(int X, int N)
{
    // Counter variable to keep the
    // count of valid numbers
    int counter = 0;
 
    for (int i = 1; counter < N; ++i)
    {
 
        // Find digital root
        int digitalRoot = findDigitalRoot(i);
 
        // Check if is required answer or not
        if (digitalRoot == X)
        {
            ++counter;
        }
 
        // Print the answer if you have found it
        // and breakout of the loop
        if (counter == N)
        {
            System.out.print( i);
 
            break;
        }
    }
}
 
// Driver Code
public static void main(String args[])
{
    int X = 1, N = 3;
 
    findAnswer(X, N);
}
}
 
// This code is contributed by Arnab Kundu

Python3




# Python3 program to find the N-th number whose
# digital root is X
import sys
 
# Function to find the digital root of
# a number
def findDigitalRoot(num):
    sum = sys.maxsize;
    tempNum = num;
 
    while (sum >= 10):
        sum = 0;
 
        while (tempNum > 0):
            sum += tempNum % 10;
            tempNum //= 10;
 
        tempNum = sum;
 
    return sum;
 
# Function to find the Nth number with
# digital root as X
def findAnswer(X, N):
     
    # Counter variable to keep the
    # count of valid numbers
    counter = 0;
    i = 0;
    while (counter < N):
        i += 1;
         
        # Find digital root
        digitalRoot = findDigitalRoot(i);
         
        # Check if is required answer or not
        if (digitalRoot == X):
            counter += 1;
         
        # Print the answer if you have found it
        # and breakout of the loop
        if (counter == N):
            print(i);
            break;
 
# Driver Code
if __name__ == '__main__':
    X = 1;
    N = 3;
 
    findAnswer(X, N);
 
# This code is contributed by 29AjayKumar

C#




// C# program to find the N-th number whose
// digital root is X
using System;
 
class GFG
{
 
// Function to find the digital root of
// a number
static int findDigitalRoot(int num)
{
    int sum = int.MaxValue, tempNum = num;
 
    while (sum >= 10)
    {
        sum = 0;
 
        while (tempNum > 0)
        {
            sum += tempNum % 10;
            tempNum /= 10;
        }
 
        tempNum = sum;
    }
 
    return sum;
}
 
// Function to find the Nth number with
// digital root as X
static void findAnswer(int X, int N)
{
    // Counter variable to keep the
    // count of valid numbers
    int counter = 0;
 
    for (int i = 1; counter < N; ++i)
    {
 
        // Find digital root
        int digitalRoot = findDigitalRoot(i);
 
        // Check if is required answer or not
        if (digitalRoot == X)
        {
            ++counter;
        }
 
        // Print the answer if you have found it
        // and breakout of the loop
        if (counter == N)
        {
            Console.Write( i);
 
            break;
        }
    }
}
 
// Driver Code
public static void Main(String []args)
{
    int X = 1, N = 3;
 
    findAnswer(X, N);
}
}
 
// This code has been contributed by 29AjayKumar

PHP




<?php
// PHP program to find the N-th number
// whose digital root is X
 
// Function to find the digital root
// of a number
function findDigitalRoot($num)
{
    $sum = PHP_INT_MAX;
    $tempNum = $num;
 
    while ($sum >= 10)
    {
        $sum = 0;
 
        while ($tempNum > 0)
        {
            $sum += $tempNum % 10;
            $tempNum /= 10;
        }
 
        $tempNum = $sum;
    }
 
    return $sum;
}
 
// Function to find the Nth number 
// with digital root as X
function findAnswer($X, $N)
{
     
    // Counter variable to keep the
    // count of valid numbers
    $counter = 0;
 
    for ($i = 1; $counter < $N; ++$i)
    {
 
        // Find digital root
        $digitalRoot = findDigitalRoot($i);
 
        // Check if is required answer or not
        if ($digitalRoot == $X)
        {
            ++$counter;
        }
 
        // Print the answer if you have found
        // it and breakout of the loop
        if ($counter == $N)
        {
            echo( $i);
 
            break;
        }
    }
}
 
// Driver Code
$X = 1; $N = 3;
 
findAnswer($X, $N);
 
// This code is contributed by Code_Mech.

Javascript




<script>
 
// JavaScript program to find the N-th number whose
// digital root is X    
 
// Function to find the digital root of
// a number
    function findDigitalRoot(num) {
        var sum = Number.MAX_VALUE, tempNum = num;
 
        while (sum >= 10) {
            sum = 0;
 
            while (tempNum > 0) {
                sum += tempNum % 10;
                tempNum = parseInt(tempNum/10);
            }
 
            tempNum = sum;
        }
 
        return sum;
    }
 
    // Function to find the Nth number with
    // digital root as X
    function findAnswer(X , N) {
        // Counter variable to keep the
        // count of valid numbers
        var counter = 0;
 
        for (var i = 1; counter < N; ++i) {
 
            // Find digital root
            var digitalRoot = findDigitalRoot(i);
 
            // Check if is required answer or not
            if (digitalRoot == X) {
                counter+=1;
            }
 
            // Print the answer if you have found it
            // and breakout of the loop
            if (counter == N) {
                document.write(i);
 
                break;
            }
        }
    }
 
    // Driver Code
     
        var X = 1, N = 3;
 
        findAnswer(X, N);
 
// This code contributed by gauravrajput1
 
</script>
Output: 
19

 

Efficient Approach: We can find digital root of a number K directly using the formula: 

digitalRoot(k) = (k - 1)mod 9 +1

From this we can find the N-th number whose digital root is K as,  

Nth number = (N - 1)*9 + K

Below is the implementation of the above approach:  

C++




// C++ program to find the N-th number with
// digital root as X
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the N-th number with
// digital root as X
int findAnswer(int X, int N)
{
    return (N - 1) * 9 + X;
}
 
// Driver Code
int main()
{
    int X = 7, N = 43;
 
    cout << findAnswer(X, N);
 
    return 0;
}

Java




// Java program to find the N-th number with
// digital root as X
 
class GfG
{
 
    // Function to find the N-th number with
    // digital root as X
    static int findAnswer(int X, int N)
    {
        return (N - 1) * 9 + X;
    }
 
    // Driver Code
    public static void main(String[] args)
    {
        int X = 7, N = 43;
        System.out.println(findAnswer(X, N));
    }
}
 
// This code contributed by Rajput-Ji

Python3




# Python3 program to find the N-th 
# number with digital root as X
 
# Function to find the N-th number
# with digital root as X
def findAnswer(X, N):
 
    return (N - 1) * 9 + X;
 
# Driver Code
X = 7;
N = 43;
print(findAnswer(X, N));
 
# This code is contributed by mits

C#




// C# program to find the N-th number
// with digital root as X
using System;
 
class GFG
{
 
// Function to find the N-th
// number with digital root as X
static int findAnswer(int X, int N)
{
    return (N - 1) * 9 + X;
}
 
// Driver Code
public static void Main()
{
    int X = 7, N = 43;
    Console.WriteLine(findAnswer(X, N));
}
}
 
// This code contributed by Ryuga

PHP




<?php
// PHP program to find the N-th number
// with digital root as X
 
// Function to find the N-th number
// with digital root as X
function findAnswer($X, $N)
{
    return ($N - 1) * 9 + $X;
}
 
// Driver Code
$X = 7; $N = 43;
echo(findAnswer($X, $N));
 
// This code contributed
// by Code_Mech
?>

Javascript




<script>
 
// Javascript program to find the N-th number
// with digital root as X
 
// Function to find the N-th number
// with digital root as X
function findAnswer(X, N)
{
    return (N - 1) * 9 + X;
}
 
// Driver Code
let X = 7;
let N = 43;
 
document.write(findAnswer(X, N));
 
// This code is contributed by mohan1240760
 
</script>
Output: 
385

 




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