1. What Is Prime Factorisation?

Prime factorisation is the process of breaking down a composite number into its prime factors. Every composite number can be expressed as a unique product of prime numbers, which is known as the Fundamental Theorem of Arithmetic.

2. How The Trial Division Algorithm Works

The trial division algorithm systematically divides the number by prime numbers:

Algorithm Steps:

• Divide by 2 repeatedly until the number is odd
• Check divisibility by odd numbers starting from 3
• Continue up to the square root of the remaining number
• If the remaining number is greater than 1, it's prime

Example: For n = 60, the prime factors are 2² × 3 × 5

3. Importance Of Prime Factorisation

Details: Prime factorisation is fundamental in number theory, cryptography (RSA algorithm), finding greatest common divisors (GCD), least common multiples (LCM), and simplifying fractions.

4. Using The Calculator

Tips: Enter any integer greater than 1. The calculator will display the prime factors in exponential notation. For example, 12 will be shown as 2² × 3.

5. Frequently Asked Questions (FAQ)

Q1: What is a prime number?
A: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Q2: Why is prime factorisation unique?
A: According to the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorisation, regardless of the order of factors.

Q3: What is the time complexity of trial division?
A: Trial division has a time complexity of O(√n) in the worst case, making it efficient for small to medium numbers but slow for very large numbers.

Q4: Can all numbers be prime factorised?
A: All composite numbers (numbers greater than 1 that are not prime) can be prime factorised. Prime numbers are already in their prime factorised form.

Q5: What are some applications of prime factorisation?
A: Applications include cryptography, computer security, number theory research, mathematical problem solving, and algorithm design.