**π‘ Problem Formulation:** In Python, we often face the problem of finding the Least Common Multiple (LCM) of a set of numbers and then determining whether that LCM is divisible by a given prime number. For instance, given an input array [3, 4, 5] and a prime number 7, we want to find out if the LCM of the array (60) is divisible by 7 (False in this case).

## Method 1: Using Math Library and Simple Iteration

This method involves using the `math.gcd()`

function to calculate the LCM of the array elements iteratively. Next, we check if the resultant LCM is divisible by the prime number using the modulus operator.

Here’s an example:

import math def find_lcm(num1, num2): return num1 * num2 // math.gcd(num1, num2) def is_lcm_divisible_by_prime(array, prime): lcm = array[0] for element in array[1:]: lcm = find_lcm(lcm, element) return lcm % prime == 0 # Example usage array = [3, 4, 5] prime = 7 print(is_lcm_divisible_by_prime(array, prime))

Output: `False`

In this snippet, the `find_lcm()`

function calculates the LCM of two numbers, which is then iteratively used to find the LCM of all elements in the array. After obtaining the final LCM, the function `is_lcm_divisible_by_prime()`

checks its divisibility by the given prime number.

## Method 2: Using NumPy Library

For those already using NumPy for array operations, NumPy’s lcm.reduce method can be utilized to find the LCM in a more concise manner. Once the LCM is obtained, we can check for divisibility by the prime number.

Here’s an example:

import numpy as np def is_lcm_divisible_by_prime(array, prime): lcm = np.lcm.reduce(array) return lcm % prime == 0 # Example usage array = np.array([3, 4, 5]) prime = 11 print(is_lcm_divisible_by_prime(array, prime))

Output: `False`

The function `is_lcm_divisible_by_prime()`

leverages NumPy’s `lcm.reduce()`

method to find the LCM of the array elements efficiently. The result is then checked for divisibility by `prime`

.

## Method 3: Using functools.reduce with a Custom LCM Function

In this method, we use the `functools.reduce()`

function along with a custom LCM function to find the LCM of the elements in the array. Checking for divisibility by the prime number remains the same.

Here’s an example:

from functools import reduce import math def find_lcm(num1, num2): return num1 * num2 // math.gcd(num1, num2) def is_lcm_divisible_by_prime(array, prime): lcm = reduce(find_lcm, array) return lcm % prime == 0 # Example usage array = [3, 4, 5] prime = 13 print(is_lcm_divisible_by_prime(array, prime))

Output: `False`

By using `reduce()`

from the `functools`

module, we can apply the `find_lcm()`

function cumulatively to the items of `array`

to compute the LCM, which is then tested against the `prime`

with the modulus operator.

## Method 4: Using a Custom Recursive LCM Function

If you prefer not to rely on libraries, you can create a custom recursive function to compute the LCM of the array. The divisibility by the prime number can be checked in the same way as before.

Here’s an example:

def gcd(a, b): if b == 0: return a return gcd(b, a%b) def find_lcm(arr): if len(arr) == 2: return arr[0] * arr[1] // gcd(arr[0], arr[1]) return find_lcm([arr[0], find_lcm(arr[1:])]) def is_lcm_divisible_by_prime(array, prime): lcm = find_lcm(array) return lcm % prime == 0 # Example usage array = [3, 4, 5] prime = 17 print(is_lcm_divisible_by_prime(array, prime))

Output: `False`

This solution computes the LCM and the GCD using recursive functions `find_lcm()`

and `gcd()`

, respectively. It’s a standalone approach that checks the divisibility by the prime number without requiring any external library.

## Bonus One-Liner Method 5: Functional Programming with lambda

For those who enjoy the functional programming paradigm, Python allows us to combine the gcd function and reduce in a single line of code to find the LCM, followed by a prime divisibility check.

Here’s an example:

from functools import reduce import math array = [3, 4, 5] prime = 19 is_lcm_divisible_by_prime = lambda arr, p: reduce(lambda x, y: x * y // math.gcd(x, y), arr) % p == 0 print(is_lcm_divisible_by_prime(array, prime))

Output: `False`

This lambda function encapsulates LCM computation and divisibility checking in a single expression. It makes use of `reduce()`

and `math.gcd()`

in a concise, although dense, format.

## Summary/Discussion

**Method 1:**Using Math Library and Simple Iteration. Strengths: Clear and easy to understand. Weaknesses: Can be slower due to explicit iteration.**Method 2:**Using NumPy Library. Strengths: Very efficient and concise for large datasets. Weaknesses: Requires the installation of the NumPy library.**Method 3:**Using functools.reduce with a Custom LCM Function. Strengths: Elegant use of functional programming. Weaknesses: Might be less intuitive for beginners.**Method 4:**Using a Custom Recursive LCM Function. Strengths: Does not depend on any external libraries. Weaknesses: Recursive methods can be less efficient and harder to understand.**Bonus Method 5:**Functional Programming with lambda. Strengths: Extremely concise. Weaknesses: Less readability, which can make the code harder to maintain.