1. What is Prime Factorization?

Prime factorization is the process of decomposing a composite number into a product of its prime factors. Every integer greater than 1 is either prime itself or can be written as a unique product of prime numbers.

2. How Does Trial Division Work?

The trial division method systematically tests divisibility by primes up to the square root of the number:

Algorithm Steps:

• Divide by 2 repeatedly until odd
• Test divisibility by odd numbers 3, 5, 7, ... up to √n
• If remaining number > 1, it's prime

Example: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

3. Importance of Prime Factorization

Applications: Used in cryptography (RSA), number theory, finding greatest common divisors, least common multiples, and simplifying fractions.

4. Using the Calculator

Instructions: Enter any integer ≥ 2. The calculator will display both the prime factorization in exponential form and the list of prime factors.

5. Frequently Asked Questions (FAQ)

Q1: Why stop at √n in trial division?
A: If n has a factor greater than √n, it must have a corresponding factor less than √n. Testing beyond √n is redundant.

Q2: What is the fundamental theorem of arithmetic?
A: Every integer greater than 1 has a unique prime factorization, regardless of the order of factors.

Q3: How efficient is trial division?
A: For numbers up to 10¹², trial division is practical. For larger numbers, more advanced algorithms like Pollard's Rho are used.

Q4: Can prime factorization be used for cryptography?
A: Yes, the difficulty of factoring large numbers is the basis for RSA encryption security.

Q5: What are prime numbers?
A: Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.