Optimizations Of All Divisors And Prime Python

Lecture 59:-  Optimizations Of All Divisors And Prime Python

Both finding all divisors and checking for prime numbers can be optimized to reduce the number of iterations and improve the efficiency of the code.

1. Optimized All Divisors:

To find all divisors of a number, you only need to iterate up to the square root of the number. If i is a divisor of the number, then its corresponding divisor num/i will also be a divisor. This optimization reduces the number of iterations and improves the performance of the function.

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import math def find_divisors_optimized(num): divisors = [] for i in range(1, int(math.sqrt(num)) + 1): if num % i == 0: divisors.append(i) if i != num // i: # Avoid duplicates when the number is a perfect square divisors.append(num // i) return divisors # Example usage: num_to_check = 12 all_divisors = find_divisors_optimized(num_to_check) print(f"All divisors of {num_to_check}: {all_divisors}")

Output:

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All divisors of 12: [1, 12, 2, 6, 3, 4]

1. Optimized Prime Check:

To check if a number is prime, you can also iterate up to the square root of the number and use the optimized all divisors approach. If the number has divisors other than 1 and itself, it is not prime.

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import math def is_prime_optimized(num): if num <= 1: return False for i in range(2, int(math.sqrt(num)) + 1): if num % i == 0: return False return True # Example usage: num_to_check = 17 if is_prime_optimized(num_to_check): print(f"{num_to_check} is a prime number.") else: print(f"{num_to_check} is not a prime number.")

The output for this example will also be:

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17 is a prime number.

By optimizing both functions with the square root approach, you significantly reduce the number of iterations required for large numbers, leading to faster execution and better performance. These optimizations are particularly useful when dealing with a large range of numbers or when you need to check for primes or find divisors frequently in your code.