Finding sum of digits of a number until sum becomes single digit

Given a number n, we need to find the sum of its digits such that: 

If n < 10    
    digSum(n) = n
Else         
    digSum(n) = Sum(digSum(n))

Examples :

Input : 1234
Output : 1
Explanation : The sum of 1+2+3+4 = 10, 
              digSum(x) == 10
              Hence ans will be 1+0 = 1

Input : 5674
Output : 4 

A brute force approach is to sum all the digits until the sum < 10. 
Flowchart:

Below is the brute force program to find the sum. 

C++

// C++ program to find sum of
// digits of a number until
// sum becomes single digit.
#include<bits/stdc++.h>

using namespace std;

int digSum(int n)
{
    int sum = 0;

    // Loop to do sum while
    // sum is not less than
    // or equal to 9
    while(n > 0 || sum > 9)
    {
        if(n == 0)
        {
            n = sum;
            sum = 0;
        }
        sum += n % 10;
        n /= 10;
    }
    return sum;
}

// Driver program to test the above function
int main()
{
    int n = 1234;
    cout << digSum(n);
    return 0;
}

Java

// Java program to find sum of
// digits of a number until
// sum becomes single digit.
import java.util.*;

public class GfG {

    static int digSum(int n)
    {
        int sum = 0;

        // Loop to do sum while
        // sum is not less than
        // or equal to 9
        while (n > 0 || sum > 9)
        {
            if (n == 0) {
                n = sum;
                sum = 0;
            }
            sum += n % 10;
            n /= 10;
        }
        return sum;
    }

    // Driver code
    public static void main(String argc[])
    {
        int n = 1234;
        System.out.println(digSum(n));
    }
}

// This code is contributed by Gitanjali.

Python

# Python program to find sum of
# digits of a number until
# sum becomes single digit.
import math

# method to find sum of digits
# of a number until sum becomes
# single digit
def digSum( n):
    sum = 0

    while(n > 0 or sum > 9):

        if(n == 0):
            n = sum
            sum = 0

        sum += n % 10
        n /= 10

    return sum

# Driver method
n = 1234
print (digSum(n))

# This code is contributed by Gitanjali.

C#

// C# program to find sum of
// digits of a number until
// sum becomes single digit.
using System;

class GFG {

    static int digSum(int n)
    {
        int sum = 0;

        // Loop to do sum while
        // sum is not less than
        // or equal to 9
        while (n > 0 || sum > 9)
        {
            if (n == 0)
            {
                n = sum;
                sum = 0;
            }
            sum += n % 10;
            n /= 10;
        }
        return sum;
    }

    // Driver code
    public static void Main()
    {
        int n = 1234;
        Console.Write(digSum(n));
    }
}

// This code is contributed by nitin mittal

PHP

<?php
// PHP program to find sum of
// digits of a number until
// sum becomes single digit.

function digSum( $n)
{
    $sum = 0;

    // Loop to do sum while
    // sum is not less than
    // or equal to 9
    while($n > 0 || $sum > 9)
    {
        if($n == 0)
        {
            $n = $sum;
            $sum = 0;
        }
        $sum += $n % 10;
        $n = (int)$n / 10;
    }
    return $sum;
}

// Driver Code
$n = 1234;
echo digSum($n);

// This code is contributed
// by aj_36
?>

Javascript

<script>
// Javascript program to find sum of
// digits of a number until
// sum becomes single digit.
    let n = 1234;
    //Function to get sum of digits
    function getSum(n) {
        let sum = 0;
        while (n > 0 || sum > 9) {
             if(n == 0) {
                n = sum;
                sum = 0;
             }
             sum = sum + n % 10;
             n = Math.floor(n / 10);
        }
        return sum;
    }
 //function call
    document.write(getSum(n));

//This code is contributed by Surbhi Tyagi
</script>

C

// C program to find sum of
// digits of a number until
// sum becomes single digit.
#include<stdio.h>

int digSum(int n)
{
    int sum = 0;

    // Loop to do sum while
    // sum is not less than
    // or equal to 9
    while(n > 0 || sum > 9)
    {
        if(n == 0)
        {
            n = sum;
            sum = 0;
        }
        sum += n % 10;
        n /= 10;
    }
    return sum;
}

// Driver program to test the above function
int main()
{
    int n = 1234;
    printf("%d",digSum(n));
    return 0;
}

Output :

1

Time Complexity: O(log(n)).
Auxiliary Space: O(1)

So, another challenge is “Could you do it without any loop/recursion in O(1) runtime?”

YES!!
There exists a simple and elegant O(1) solution for this too. The answer is given simply:- 

If n == 0
   return 0;

If n % 9 == 0      
    digSum(n) = 9
Else               
    digSum(n) = n % 9 

How does the above logic works? 

The logic behind this approach is :

To check if a number is divisible by 9, add the digits of the number and check if the sum is divisible by 9 or not. If yes, is the case,  the number is divisible by 9, otherwise, it’s not.

let’s take 27  i.e (2+7 = 9) hence divisible by 9.
If a number n is divisible by 9, then the sum of its digit until the sum becomes a single digit is always 9. For example, 
Let, n = 2880 
Sum of digits = 2 + 8 + 8 = 18: 18 = 1 + 8 = 9

Therefore,
A number can be of the form 9x or 9x + k. For the first case, the answer is always 9. For the second case, and is always k which is the remainder left.

The problem is widely known as the digit root problem.

You may find this Wikipedia article useful. -> https://en.wikipedia.org/wiki/Digital_root

Below is the implementation of the above idea : 

C++

#include<bits/stdc++.h>
using namespace std;

int digSum(int n)
{
    if (n == 0)
       return 0;
    return (n % 9 == 0) ? 9 : (n % 9);
}

// Driver program to test the above function
int main()
{
    int n = 9999;
    cout<<digSum(n);
    return 0;
}

Java

import java.io.*;

class GFG {

    static int digSum(int n)
    {
        if (n == 0)
        return 0;
        return (n % 9 == 0) ? 9 : (n % 9);
    }

    // Driver program to test the above function
    public static void main (String[] args)
    {
        int n = 9999;
        System.out.println(digSum(n));
    }
}

// This code is contributed by anuj_67.

Python3

def digSum(n):

    if (n == 0):
        return 0
    if (n % 9 == 0):
        return 9
    else:
       return (n % 9)

# Driver program to test the above function
n = 9999
print(digSum(n))

# This code is contributed by
# Smitha Dinesh Semwal

C#

using System;

class GFG
{
    static int digSum(int n)
    {
        if (n == 0)
        return 0;
        return (n % 9 == 0) ? 9 : (n % 9);
    }

    // Driver Code
    public static void Main ()
    {
        int n = 9999;
        Console.Write(digSum(n));

    }
}

// This code is contributed by aj_36

PHP

<?php

function digSum($n)
{
    if ($n == 0)
        return 0;
    return ($n % 9 == 0) ? 9 : ($n % 9);
}

// Driver program to test the above function
$n = 9999;
echo digSum($n);

//This code is contributed by anuj_67.
?>

Javascript

<script>

function digSum(n)
{
    if (n == 0)
        return 0;

    return (n % 9 == 0) ? 9 : (n % 9);
}

// Driver code
n = 9999;
document.write(digSum(n));

// This code is contributed by code_hunt

</script>

Output:

9

Time Complexity: O(1)

Auxiliary Space: O(1)

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