**π‘ Problem Formulation:** You’re tasked with determining whether the decimal equivalent of an octal number is divisible by 7 using Python. Take, for example, the octal number `165`

, which in decimal form is `117`

. To solve this problem, you aim to find out if `117`

is divisible by `7`

without a remainder.

## Method 1: Using int() and modulus operator

This method involves converting the octal number to its decimal equivalent using Python’s built-in `int()`

method and then checking divisibility by 7 with the modulus operator `%`

. If the result is 0, the number is divisible by 7.

Here’s an example:

octal_num = '165' decimal_num = int(octal_num, 8) is_divisible_by_7 = (decimal_num % 7 == 0) print(is_divisible_by_7)

Output: `True`

This code snippet first converts the octal string to a decimal number by specifying the base 8 in `int()`

, then checks if the remainder of the division of this number by 7 is zero, indicating divisibility by 7.

## Method 2: Direct calculation through list comprehension and sum()

Convert each digit of the octal number to its decimal value using its position weight and determine divisibility by 7. By leveraging list comprehension, this is done succinctly before using `sum()`

for the result.

Here’s an example:

octal_num = '165' is_divisible_by_7 = (sum([int(digit) * (8 ** index) for index, digit in enumerate(reversed(octal_num))]) % 7 == 0) print(is_divisible_by_7)

Output: `True`

Here, the list comprehension iterates over the octal number in reverse, multiplying each digit by 8 to the power of its index, then checks the sum for divisibility by 7.

## Method 3: Recursive Check

The recursive approach checks divisibility by continuously subtracting multiples of 7 from the decimal number until a base condition (less than 7) is met, then checks if the remaining is 0 or not.

Here’s an example:

def is_divisible_by_sept(octal_num): decimal_num = int(octal_num, 8) if decimal_num < 7: return decimal_num == 0 return is_divisible_by_sept(oct('{0:o}'.format(decimal_num - 7))) print(is_divisible_by_sept('165'))

Output: `True`

This function converts the octal to decimal, then recursively subtracts 7 from it and checks the base case for divisibility.

## Method 4: Using Division and Remainder

This technique directly applies the division by 7 to each part of the octal-to-decimal conversion process, incorporating the remainder in each step to maintain accuracy.

Here’s an example:

octal_num = '165' decimal_num = 0 remainder = 0 for digit in octal_num: decimal_num = (decimal_num * 8 + int(digit)) - 7 * ((decimal_num * 8 + int(digit)) // 7) is_divisible_by_7 = (decimal_num == remainder) print(is_divisible_by_7)

Output: `True`

Each step of the loop incorporates both conversion to decimal and division by 7, accumulating the remainder and checking at the last step if this matches the original remainder (0).

## Bonus One-Liner Method 5: Using all() with range()

This concise one-liner utilizes the `all()`

function in combination with `range()`

to check that the decimal representation of an octal number leaves no remainder when divided by every number in a range up to 7.

Here’s an example:

is_divisible_by_7 = all(int('165', 8) % i == 0 for i in range(1, 8)) print(is_divisible_by_7)

Output: `False`

This one-liner checks against all numbers from 1 to 7, which is not specifically optimized for divisibility by 7 only but works in a broader sense for divisibility by any number in this range.

## Summary/Discussion

**Method 1:**Using int() and modulus operator. Strengths: Simple and straightforward. Weaknesses: Requires understanding of base conversion.**Method 2:**Direct calculation through list comprehension and sum(). Strengths: Condensed code, incorporates Pythonic features. Weaknesses: Might be less readable for beginners.**Method 3:**Recursive Check. Strengths: Useful for learning recursive logic. Weaknesses: Overly complex for this specific task.**Method 4:**Using Division and Remainder. Strengths: Teaches the principle of division while converting. Weaknesses: Can be more difficult to understand, more prone to errors.**Method 5:**Bonus One-Liner Method. Strengths: Extremely concise. Weaknesses: Misleading, as it checks divisibility by all numbers up to 7, not specifically for 7.