while (n > 1) {
    for (int i = 2; i <= sqrt(n); i++) {
        while (n % i == 0) {
            // i is a prime factor
            n /= i;
        }
    }
    if (n > 1) {
        // n is prime
    }
}
    

1. What Are Prime Factors?

Prime factors are the prime numbers that divide a given integer exactly, without leaving a remainder. Every integer greater than 1 can be expressed as a unique product of prime numbers.

2. How Does the Algorithm Work?

The algorithm uses trial division to find prime factors:

Algorithm Steps:
1. Divide by 2 repeatedly while n is even
2. Check odd divisors from 3 up to √n
3. For each divisor i, divide n by i while divisible
4. If remaining n > 1, it's a prime factor
                    

Time Complexity: O(√n) in worst case

Space Complexity: O(k) where k is number of prime factors

3. Importance of Prime Factorization

Applications: Prime factorization is fundamental in cryptography (RSA), number theory, computer science algorithms, and mathematical problem solving.

4. Using the Calculator

Instructions: Enter any integer greater than 1. The calculator will display the prime factorization in expanded form (e.g., 12 = 2 × 2 × 3).

5. Frequently Asked Questions (FAQ)

Q1: What is the fundamental theorem of arithmetic?
A: It states that every integer greater than 1 can be represented exactly one way as a product of prime numbers.

Q2: Why check only up to √n?
A: If n has a factor greater than √n, it must have a corresponding factor less than √n, so checking up to √n is sufficient.

Q3: What are some efficient prime factorization methods?
A: For very large numbers, Pollard's Rho, Quadratic Sieve, or General Number Field Sieve are more efficient than trial division.

Q4: Can prime factors include 1?
A: No, 1 is not considered a prime number. Prime factors must be prime numbers greater than 1.

Q5: What is the prime factorization of prime numbers?
A: Prime numbers are their own prime factorization (e.g., 7 = 7).