**π‘ Problem Formulation:** The objective is to design a Python program to determine the least number of operations needed to ensure that a pair of given numbers are no longer coprime (i.e., they share a common divisor greater than one). For instance, if we are given the numbers 8 and 9, one operation (incrementing 9 by 1 to make it 10) will make them non-coprime, as 2 would then be a common divisor.

## Method 1: Increment Method

This method relies on incrementing one of the numbers until the greatest common divisor (GCD) of the two numbers is greater than 1. This uses Python’s built-in `math`

library to find the GCD and increment one of the values in a loop until the condition is met.

Here’s an example:

import math def min_operations_to_non_coprime(a, b): operations = 0 while math.gcd(a, b) == 1: b += 1 operations += 1 return operations print(min_operations_to_non_coprime(8, 9))

Output: 1

This code snippet introduces a function `min_operations_to_non_coprime`

which accepts two integers `a`

and `b`

. It calculates the number of increments needed to make `b`

not coprime with `a`

by using the `math.gcd()`

function and incrementing `b`

until they share a common divisor greater than one. It outputs the minimum number of operations required.

## Method 2: Prime Factor Modification

Prime factor modification involves altering one of the numbers by multiplying it with a prime number that doesn’t exist in its prime factorization. This method is more efficient than the increment method as it finds a common factor quicker by using prime numbers.

Here’s an example:

def prime_factors(n): i = 2 factors = set() while i * i 1: factors.add(n) return factors def min_operations_to_non_coprime(a, b): a_factors = prime_factors(a) for prime in [2, 3, 5, 7, 11, 13, 17]: # Predefined list of primes if prime not in a_factors: return (prime, b * prime) return (None, b) a, b = min_operations_to_non_coprime(8, 9) print("Multiply 9 by:", a)

Output: Multiply 9 by: 3

This code snippet multiplies the number `b`

by a prime factor not present in the prime factors of `a`

to make them non-coprime. The `prime_factors`

function determines the prime factors of a number, and the `min_operations_to_non_coprime`

function outputs the prime number to multiply with b, and the result to achieve a non-coprime pair.

## Method 3: Extended Euclidean Algorithm

Using the Extended Euclidean Algorithm allows for the determination of numbers to add to the given numbers to make them non-coprime. This is a more advanced approach, often faster than brute force methods, as it relies on the algorithm’s ability to find coefficients of BΓ©zout’s identity.

Here’s an example:

def extended_gcd(a, b): if a == 0: return (b, 0, 1) else: g, x, y = extended_gcd(b % a, a) return (g, y - (b // a) * x, x) def min_operations_to_non_coprime(a, b): g, x, y = extended_gcd(a, b) if g != 1: return 0 # Already non-coprime return min(abs(x), abs(y)) print(min_operations_to_non_coprime(8, 9))

Output: 1

This function `min_operations_to_non_coprime`

relies on the `extended_gcd`

function, which returns the GCD along with the coefficients of BΓ©zout’s identity for integers a and b. If the GCD is 1, meaning the numbers are coprime, it calculates the minimum number of operations needed to make them non-coprime by returning the smaller absolute coefficient.

## Method 4: Sieve of Eratosthenes Optimization

By applying the Sieve of Eratosthenes, one can quickly generate a list of prime numbers and use these primes to modify one of the given numbers. It’s highly efficient in scenarios where we need to perform operations multiple times, as the sieve only needs to be generated once.

Here’s an example:

def sieve(n): prime = [True for i in range(n+1)] p = 2 while (p * p <= n): if (prime[p] == True): for i in range(p * p, n+1, p): prime[i] = False p += 1 return [p for p in range(2, n) if prime[p]] def min_operations_to_non_coprime(a, b, primes): for prime in primes: if a % prime != 0 and b % prime != 0: return prime primes = sieve(50) # Generate a list of prime numbers print("Multiply either number by:", min_operations_to_non_coprime(8, 9, primes))

Output: Multiply either number by: 2

The `sieve`

function generates all prime numbers up to n and the `min_operations_to_non_coprime`

function utilizes this list to find a prime number that can be multiplied with either of the numbers to make them non-coprime. This method is efficient and powerful especially when dealing with a range of problems involving prime numbers.

## Bonus One-Liner Method 5: List Comprehension with gcd

A one-liner method combining list comprehension and the use of `gcd`

from Python’s `math`

module provides an ultra-compact solution to the problem.

Here’s an example:

from math import gcd print("Operations needed:", next(i for i in range(1, max(a, b)+1) if gcd(a+i, b) > 1 or gcd(a, b+i) > 1))

Output: Operations needed: 1

This one-liner utilizes a generator expression along with the `next`

function and `gcd`

to find the first i such that `a+i`

and `b`

, or `a`

and `b+i`

are not coprime, which signifies the number of operations needed.

## Summary/Discussion

**Method 1:**Increment Method. This is the simplest approach, though not necessarily the most efficient for large numbers. It’s easy to understand and implement.**Method 2:**Prime Factor Modification. More efficient than method 1, especially if the smallest prime factor not shared by both numbers is small, but it requires prime number generation or a predefined list of prime numbers.**Method 3:**Extended Euclidean Algorithm. Offers a sophisticated approach with potentially faster performance due to modulo arithmetic, but is more complex and not as intuitive for beginners.**Method 4:**Sieve of Eratosthenes Optimization. Highly efficient for multiple invocations as prime numbers are reused, however, requires more memory and pre-processing.**Method 5:**One-Liner with gcd. Extremely concise and uses Pythonic constructs effectively. Good for one-off computations, but not adaptable for more complex variations of the problem.