How to Calculate Prime Factorisation: Step-by-Step Guide

Prime factorisation is a process of breaking down a composite number into a product of prime numbers. This guide will walk you through the steps to calculate prime factorisation manually.

Introduction to Prime Factorisation

Prime factorisation is a fundamental concept in number theory, and it has numerous applications in mathematics and computer science. The formula for prime factorisation is: N = p1^a1 * p2^a2 * ... * pn^an, where N is the number to be factorised, p1, p2, ..., pn are prime numbers, and a1, a2, ..., an are their corresponding exponents.

Step-by-Step Guide

The steps to calculate prime factorisation are as follows:

Step 1: Understand the Concept of Prime Numbers

Before starting the prime factorisation process, it's essential to understand what prime numbers are. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Examples of prime numbers are 2, 3, 5, 7, 11, etc.

Step 2: Find the Smallest Prime Factor

To start the prime factorisation process, find the smallest prime number that divides the given number. This can be done by trial division or using a factor tree. For example, if we want to find the prime factorisation of 24, we start by dividing it by the smallest prime number, which is 2.

Step 3: Divide and Repeat

Once we find a prime factor, we divide the number by that factor and repeat the process until we cannot divide anymore. Using the same example as above, 24 ÷ 2 = 12. Then, 12 ÷ 2 = 6, 6 ÷ 2 = 3. Since 3 is a prime number, we cannot divide further.

Step 4: Write the Prime Factorisation in Exponent Notation

After finding all the prime factors, we write the prime factorisation in exponent notation. Using the same example, the prime factorisation of 24 is 2^3 * 3^1.

Worked Example

Let's find the prime factorisation of 48.

• Start by dividing 48 by the smallest prime number, which is 2: 48 ÷ 2 = 24
• Continue dividing by 2 until we cannot divide anymore: 24 ÷ 2 = 12, 12 ÷ 2 = 6, 6 ÷ 2 = 3
• Since 3 is a prime number, we cannot divide further
• The prime factorisation of 48 is 2^4 * 3^1

Common Mistakes to Avoid

When calculating prime factorisation manually, it's essential to avoid the following common mistakes:

• Forgetting to check for divisibility by the smallest prime numbers
• Not repeating the division process until we cannot divide anymore
• Not writing the prime factorisation in exponent notation

When to Use a Calculator

While it's essential to learn how to calculate prime factorisation manually, there are situations where using a calculator is more convenient. For large numbers, using a calculator can save time and reduce the chance of errors. Additionally, calculators can be used to verify our manual calculations.