**π‘ Problem Formulation:** The challenge is to construct a program in Python that accepts a number and a base (k), and returns the sum of its digits when the number is expressed in base k. For instance, if the input number is 345 and base k is 8, the number in base 8 is 531, and the sum of digits in base k would be 5+3+1=9.

## Method 1: Basic Iterative Approach

This method involves converting the given number into its base k equivalent and then summing its digits. This is a straightforward technique that uses while loops and modular arithmetic to achieve the conversion and summation effectively.

Here’s an example:

def sum_of_digits_base_k(number, k): base_k_number = '' while number > 0: base_k_number = str(number % k) + base_k_number number = number // k return sum(int(digit) for digit in base_k_number) # Example usage: print(sum_of_digits_base_k(345, 8))

**Output:** 9

This code snippet defines a function `sum_of_digits_base_k`

, which first converts the number from base 10 to base k. It then sums the digits of the base k number using a list comprehension and returns the result. It is an intuitive method for anyone familiar with the base conversion process.

## Method 2: Using Built-in Functions

Python’s built-in functions `divmod()`

and `int()`

can be utilized to simplify the iterative method. This provides a more Pythonic and concise solution.

Here’s an example:

def sum_of_digits_base_k(number, k): sum_of_digits = 0 while number: number, digit = divmod(number, k) sum_of_digits += digit return sum_of_digits # Example usage: print(sum_of_digits_base_k(345, 8))

**Output:** 9

In this code snippet, the function `sum_of_digits_base_k`

uses the `divmod()`

function to simultaneously perform division and modulo operations in a single step, iteratively reducing the number and accumulating the sum of its base k digits.

## Method 3: Recursive Approach

A recursive approach can be elegant and conceptually simple. It reduces the problem into smaller subproblems until a base case is reached, then combines the results.

Here’s an example:

def sum_of_digits_base_k(number, k): if number < k: return number else: return number % k + sum_of_digits_base_k(number // k, k) # Example usage: print(sum_of_digits_base_k(345, 8))

**Output:** 9

This snippet demonstrates recursion in the function `sum_of_digits_base_k`

, which calls itself with a reduced problem size until the base case is reached. This approach is shorter but can be less intuitive to understand and may result in a stack overflow for large numbers.

## Method 4: Python Libraries for Base Conversion

There are Python libraries that simplify base conversionβthus allowing a direct summation of digits without the need for a custom conversion function.

Here’s an example:

import numpy as np def sum_of_digits_base_k(number, k): base_k_number = np.base_repr(number, base=k) return sum(int(digit) for digit in base_k_number) # Example usage: print(sum_of_digits_base_k(345, 8))

**Output:** 9

In this approach, the `numpy`

library’s function `np.base_repr()`

is utilized to handle base conversions effortlessly. Then, the sum of the converted number’s digits is calculated. This method is highly readable, but it requires an external library.

## Bonus One-Liner Method 5: Using Intuitive Python Features

This involves leveraging Python’s powerful language features to fit the functionality into a single line, making it extremely concise.

Here’s an example:

print(sum(int(digit) for digit in np.base_repr(345, base=8)))

**Output:** 9

The one-liner leverages comprehension and the `numpy`

library to perform the conversion and sum of digits all in one succinct line of code, showcasing the power of Python’s expressive syntax.

## Summary/Discussion

**Method 1:**Basic Iterative Approach. Simple and understandable. Can be slow for very large numbers.**Method 2:**Using Built-in Functions. More Pythonic and concise. Still iterative, so potentially slow for huge numbers.**Method 3:**Recursive Approach. Elegant, short code. Risk of stack overflow for very large numbers.**Method 4:**Python Libraries. High readability and ease of use. Requires external dependency on`numpy`

library.**Method 5:**One-Liner. Ultra-concise and showcases Python’s strengths. Depends on`numpy`

, and readability might suffer for new coders.