Python Program for Find largest prime factor of a number

Given a positive integer \'n\'( 1 <= n <= 10¹⁵). Find the largest prime factor of a number.

Input: 6
Output: 3
Explanation
Prime factor of 6 are- 2, 3
Largest of them is 3

Input: 15
Output: 5

# Python3 code to find largest prime
# factor of number
import math

# A function to find largest prime factor
def maxPrimeFactors (n):
    
    # Initialize the maximum prime factor
    # variable with the lowest one
    maxPrime = -1
    
    # Print the number of 2s that divide n
    while n % 2 == 0:
        maxPrime = 2
        n >>= 1     # equivalent to n /= 2
        
    # n must be odd at this point, 
    # thus skip the even numbers and 
    # iterate only for odd integers
    for i in range(3, int(math.sqrt(n)) + 1, 2):
        while n % i == 0:
            maxPrime = i
            n = n / i
    
    # This condition is to handle the 
    # case when n is a prime number 
    # greater than 2
    if n > 2:
        maxPrime = n
    
    return int(maxPrime)

# Driver code to test above function
n = 15
print(maxPrimeFactors(n))

n = 25698751364526
print(maxPrimeFactors(n))

# This code is contributed by "Sharad_Bhardwaj".

5
328513

Time complexity: \text{O}(\sqrt{n})     
Auxiliary space: \text{O}(1)     
Please refer complete article on Find largest prime factor of a number for more details!

Method 2:

Follow the steps below for the implementation:

• Initialize variables largest_prime to -1, i to 2, and n to the input integer.
• Start a while loop that continues as long as i * i <= n. This loop will iterate through all possible factors of n.
• In the while loop, start another while loop that continues as long as n % i == 0. This inner loop will divide n by i until n is no longer divisible by i.
• In the inner loop, set largest_prime to i, and update n by dividing it by i.
• At the end of the inner loop, increment i by 1.
• After the outer loop, if n > 1, set largest_prime to n. This is because n could be a prime number larger than any of its factors.
• Return largest_prime.

def largest_prime_factor(n):
    """
    Find the largest prime factor of a positive integer 'n'
    :param n: positive integer ( 1 <= n <= 10^15)
    :return: largest prime factor of n
    """
    largest_prime = -1
    i = 2
    while i * i <= n:
        while n % i == 0:
            largest_prime = i
            n = n // i
        i = i + 1
    if n > 1:
        largest_prime = n
    return largest_prime

n = 15
print(largest_prime_factor(n))

n = 25698751364526
print(largest_prime_factor(n))

5
328513

Time complexity:  O(sqrt(n)).
Auxiliary space: O(1),