Python Program to Find Largest Prime Factor of a Number

Given a positive integer n (1 ≤ n ≤ 10¹⁵), the task is to find its largest prime factor the biggest prime number that divides n exactly. For Example:

Input: n = 6
Output: 3
Explanation: prime factors of 6 are 2 and 3, and the largest among them is 3.

Let's explore different methods to find largest prime factor of a number in Python.

Using factorint()

sympy provides the function factorint() which directly returns all prime factors of a number and their powers, and using max() we can extract the largest number.

from sympy import factorint
n = 15
factors = factorint(n)   
print(max(factors))      

Output

5

Using primefactors()

Another sympy method is primefactors(), it directly gives a sorted list of all prime factors of a number.

from sympy import primefactors
n = 18
print(max(primefactors(n)))

Output

3

Using math Module

This method divides the number continuously by small divisors starting from 2 until only the largest prime factor remains.

import math
n = 34
max_prime = -1

while n % 2 == 0:
    max_prime = 2
    n //= 2

for i in range(3, int(math.sqrt(n)) + 1, 2):
    while n % i == 0:
        max_prime = i
        n //= i

if n > 2:
    max_prime = n
print(max_prime)

17

Explanation:

• math.sqrt() limits checks up to √n for efficiency.
• Removes all factors of 2 first, then tests odd divisors.
• Each valid divisor updates max_prime.
• Any leftover number > 2 is the largest prime factor.

Using Iterative Division

This method does not use any external modules. It uses a loop that keeps dividing the number by its smallest divisor until all smaller factors are removed.

n = 45
i = 2
res = -1

while i * i <= n:
    while n % i == 0:
        res = i
        n //= i
    i += 1

if n > 1:
    res = n

print(res)

5

Explanation:

• The loop runs while i*i is less than or equal to n, ensuring all possible factors are checked.
• When a divisor divides n, it is recorded and removed completely.
• The loop increments i to check for the next factor.
• After all smaller factors are removed, the leftover number (if greater than 1) is the largest prime factor.