Add Digits

Problem Statement

When you encounter a problem that involves reducing a number down to a single-digit summary or identity, digital root is often behind the scenes. In the given prompt, the task is to compute the digital root of a non-negative integer num. This is accomplished by repeatedly summing the digits of num until only one digit remains. The operation involves breaking down the number into its component digits, summing them, and repeating the process with the new sum until it's a single digit.

The input, num, ranges from 0 up to 2^31 - 1, which sets a wide range for potential input values. For instance, starting with num = 38, the digits of 38 (3 + 8) sum to 11. Subsequently, adding the digits of 11 (1 + 1) results in 2, and since 2 is a single digit, it's the output.

The minimal nature of this problem inherently relies on both the properties of numbers and a bit of looping logic or recursion to achieve the result.

Examples

Example 1

Input:

num = 38

Output:

2

Explanation:

The process is
38 --> 3 + 8 --> 11
11 --> 1 + 1 --> 2
Since 2 has only one digit, return it.

Example 2

Input:

num = 0

Output:

0

Constraints

• 0 <= num <= 231 - 1

Approach and Intuition

This problem can be approached directly as described or through the properties of number theory, specifically utilizing the concept of digital roots and modulo operation. The key observations and possible implementation steps can be broken down as follows:

1. Direct Summation Method:

   • Initialize a variable to store the current sum of digits.
   • While the number has more than one digit (i.e., 10 or higher):
     • Break down the number into its individual digits.
     • Sum these digits and assign the result as the new number.
     • Repeat until the number is a single digit.
   • Return the resulting single-digit number.

2. Optimized Digital Root Method Using Modulo:

   • Recognize that the process of summing the digits repeatedly can be related to finding the digital root through modulo operations.
   • Mathematical observation tells us that the digital root of a number is equivalent to that number modulo 9, except when the number is 0 or a multiple of 9.
   • Use the formula: num % 9 and special case handling such as:
     • If num equals zero, return 0 immediately as its digital root.
     • If num % 9 equals zero (and num is not zero), return 9.
     • Otherwise, return num % 9.

The modulo approach is particularly efficient as it slices down the repetitive manual summation work seen in the direct summation method and reaches the answer in constant time, making it suitable for large numbers. However, it's important to treat both methods, as each holds distinct learning opportunities and insights into processing numbers and their properties.

Solutions

class Solution {
public:
    int sumDigits(int n) {
        if (n == 0) return 0;
        if (n % 9 == 0) return 9;
        return n % 9;
    }
};

class Solution {

    public int computeSumOfDigits(int value) {
        if (value == 0) return 0;
        if (value % 9 == 0) return 9;
        return value % 9;
    }
}

class Solution:
    def sumDigits(self, number: int) -> int:
        if number == 0:
            return 0
        if number % 9 == 0:
            return 9
        return number % 9