Digital Root (repeated digital sum) of the given large integer

Last Updated : 18 Apr, 2025

The digital root of a positive integer is found by summing the digits of the integer. If the resulting value is a single digit then that digit is the digital root. If the resulting value contains two or more digits, those digits are summed and the process is repeated. This is continued as long as necessary to obtain a single digit.
Given a number, the task is to find its digital root. The input number may be large and it may not be possible to store even if we use long long int.

Examples :

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

Input : num = “5674”
Output : 4

[Naive Approach] By Repetitively Adding Digits

The approach is focused on calculating the digital root of a number, which is the result of summing the digits repeatedly until a single-digit value is obtained. Here’s how it works conceptually:

Sum the digits: Start by adding all the digits of the given number.
Check the result: If the sum is a single-digit number (i.e., less than 10), stop and return it.
Repeat the process: If the sum is still more than a single digit, repeat the process with the sum of digits. This continues until a single-digit sum is reached.

Example:

For a number like 123

First, sum the digits: 1 + 2 + 3 + 4 = 10.
Since 10 is not a single digit, sum the digits of 10: 1 + 0 = 1.
Now, 1 is a single digit, so we stop and return 1.

To read about code implementation Refer, Finding the sum of digits of a number until the sum becomes a single digit

[Expected Approach] Using Mathematical Formula

We know that every number in the decimal system can be expressed as a sum of its digits multiplied by powers of 10. For example, a number represented as abcd can be written as follows:

abcd = a*10^3 + b*10^2 + c*10^1 + d*10^0

We can separate the digits and rewrite this as:
abcd = a + b + c + d + (a*999 + b*99 + c*9)
abcd = a + b + c + d + 9*(a*111 + b*11 + c)

This implies that any number can be expressed as the sum of its digits plus a multiple of 9.
So, if we take modulo with 9 on each side,
abcd % 9 = (a + b + c + d) % 9 + 0

This means that the remainder when abcd is divided by 9 is equal to the remainder where the sum of its digits (a + b + c + d) is divided by 9.

If the sum of the digits itself consists of more than one digit, we can further express this sum as the sum of its digits plus a multiple of 9. Consequently, taking modulo 9 will eliminate the multiple of 9, until the sum of digits become single digit number.

As a result, the sum of the digits of any number, will equal its modulo 9. If the result of the modulo operation is zero, it indicates that the single-digit result is 9.

C++

// C++ program to find the digit sum  // using mathematical formula  #include <iostream> using namespace std;  int singleDigit(int n) {          // If given number is zero its       // digit sum will be zero only     if (n == 0)         return 0;            // If result of modulo operation is        // zero then, the digit sum is 9       if(n % 9 == 0)           return 9;             return (n % 9); }  int main() {     int n = 1234;     cout << singleDigit(n);     return 0; }

// C program to find the digit sum  // using mathematical formula  #include <stdio.h>  int singleDigit(int n) {          // If given number is zero its       // digit sum will be zero only     if (n == 0)         return 0;            // If result of modulo operation is        // zero then, the digit sum is 9       if(n % 9 == 0)           return 9;             return (n % 9); }  int main() {     int n = 1234;     printf("%d", singleDigit(n));     return 0; }

Java

// Java program to find the digit sum  // using mathematical formula  class GfG {     static int singleDigit(int n) {            // If given number is zero its         // digit sum will be zero only         if (n == 0)              return 0;              // If result of modulo operation is          // zero then, the digit sum is 9         if(n % 9 == 0)             return 9;               return (n % 9);     }      public static void main(String[] args) {         int n = 1234;         System.out.println(singleDigit(n));     } }

Python

# Python program to find the digit sum  # using mathematical formula  def singleDigit(n):        # If given number is zero its     # digit sum will be zero only     if n == 0:          return 0          # If result of modulo operation is      # zero then, the digit sum is 9     if n % 9 == 0:         return 9          return n % 9  if __name__ == "__main__":     n = 1234     print(singleDigit(n))

// C# program to find the digit sum  // using mathematical formula  using System;  class GfG {     static int singleDigit(int n) {            // If given number is zero its         // digit sum will be zero only         if (n == 0)              return 0;              // If result of modulo operation is          // zero then, the digit sum is 9         if(n % 9 == 0)             return 9;               return (n % 9);     }      static void Main() {         int n = 1234;         Console.WriteLine(singleDigit(n));     } }

JavaScript

// JavaScript program to find the digit sum  // using mathematical formula  function singleDigit(n) {        // If given number is zero its     // digit sum will be zero only     if (n === 0)          return 0;          // If result of modulo operation is      // zero then, the digit sum is 9     if(n % 9 === 0)         return 9;          return n % 9; }  // Driver Code const n = 1234; console.log(singleDigit(n));