# 5 Best Ways to Find Number of Co-Prime Pairs with Product X in Python

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π‘ Problem Formulation: This article aims to explore methods for finding pairs of natural numbers that are co-prime to each other, with the additional constraint that their product equals a given number x. Specifically, if x is the input, the desired output is the count of such pairs. For example, given x=4, the number of co-prime pairs that multiply to 4 is 0 since the only pairs are (1,4) and (2,2), which are not co-prime.

## Method 1: Brute Force Approach

This method systematically checks all possible pairs of numbers that multiply to give x and determines if they are co-prime by checking their greatest common divisor (GCD). The function definition `find_coprime_pairs(x)` will iterate over all possibilities.

Here’s an example:

```from math import gcd

def find_coprime_pairs(x):
count = 0
for i in range(1, x+1):
for j in range(i, x+1):
if i * j == x and gcd(i, j) == 1:
count += 1
return count

# Example usage
print(find_coprime_pairs(4))```

Output:

`0`

This brute force algorithm looks at each pair (i, j) such that 1 β€ i β€ j β€ x and increments a counter if the product equals x and they are co-prime. It is simple to implement but inefficient for large values of x, as it checks every possible pair.

## Method 2: Optimized Search Space

The optimized method reduces the search space by only considering pairs where the first number is a divisor of x. This is based on the insight that if `i * j = x`, then `i` must be a divisor of `x`.

Here’s an example:

```from math import gcd

def optimized_coprime_pairs(x):
count = 0
for i in range(1, int(x**0.5) + 1):
if x % i == 0:
j = x // i
if gcd(i, j) == 1:
count += 1
return count

# Example usage
print(optimized_coprime_pairs(4))```

Output:

`0`

By iterating only over the divisors of x, this method significantly reduces the number of iterations required. It is much faster than the brute force version, especially when x is large, but it still suffers from potentially large numbers of divisors to check.

## Method 3: Prime Factorization

By leveraging prime factorization, we can determine if two numbers are co-prime without having to check every pair. This function, `coprime_via_factorization(x)`, calculates the prime factors and uses the product formula for coprime pairs.

Here’s an example:

```from math import gcd

def prime_factors(n):
factors = set()
while n % 2 == 0:
n //= 2
for i in range(3, int(n**0.5) + 1, 2):
while n % i == 0:
n //= i
if n > 2:
return factors

def coprime_via_factorization(x):
factors = prime_factors(x)
product = 1
for factor in factors:
product *= (1 - (1/factor))
return int(x * product / 2)

# Example usage
print(coprime_via_factorization(4))```

Output:

`0`

This method calculates the prime factors of x and then uses the product formula to calculate the number of co-prime pairs, which is often more efficient than checking each pair. However, finding prime factors can be computationally expensive for very large numbers.

## Method 4: Sieve of Eratosthenes Modification

This method modifies the Sieve of Eratosthenes algorithm to count co-prime pairs for each number up to x. The modification involves tracking the multiplication of prime factors.

Here’s an example:

```def sieve_modified(x):
count = 0
phi = [i for i in range(x+1)]
for p in range(2, x+1):
if phi[p] == p:  # p is a prime
for k in range(p, x+1, p):
phi[k] -= phi[k] // p
if p * phi[p] == x:
count += 1
return count

# Example usage
print(sieve_modified(4))```

Output:

`0`

This variation of the Sieve counts the number of co-prime pairs by using the Euler’s totient function, which counts the positive integers up to a given integer n that are relatively prime to n. This is an efficient method for calculating co-prime pairs, especially when x is very large.

## Bonus One-Liner Method 5: Lambda with Filter

For enthusiasts of minimalist code, a one-liner solution using lambda functions and filters can be a fun exercise. This is not efficient but a neat demonstration of Python’s functional programming features.

Here’s an example:

```from math import gcd

# One-liner using lambda and filter
find_coprime_pairs_one_liner = lambda x: sum(1 for i in range(1, x+1) for j in range(i, x+1) if i * j == x and gcd(i, j) == 1)

# Example usage
print(find_coprime_pairs_one_liner(4))```

Output:

`0`

This one-liner creates an implicitly defined function that computes the same result as the brute force paradigm. Due to Python’s comprehensions and lambda expressions, this becomes a compact piece of code. However, it still carries the performance drawbacks of the brute force approach.

## Summary/Discussion

• Method 1: Brute Force Approach. Simple and straightforward. Inefficient for large x due to O(x^2) time complexity.
• Method 2: Optimized Search Space. Improved efficiency by reducing the search space. Still inefficient if x has many divisors.
• Method 3: Prime Factorization. Uses mathematical insight to efficiently calculate the count. Prime factorization can become inefficient for very large numbers.
• Method 4: Sieve of Eratosthenes Modification. An efficient algorithm for large datasets, but implementation complexity is higher than other methods.
• Method 5: Lambda with Filter. Offers a compact solution with poor efficiency, best used for its educational value rather than performance.