What Are The Prime Factors Of 225

What are the Prime Factors of 225? A Step-by-Step Guide to Prime Factorization

Finding the prime factors of a number is a fundamental concept in mathematics, particularly in number theory and algebra. This guide will walk you through the process of determining the prime factors of 225, explaining the method and concepts involved. Understanding prime factorization is crucial for various mathematical operations, including simplifying fractions and solving equations.

Understanding Prime Numbers and Prime Factorization

Before we dive into finding the prime factors of 225, let's define some key terms:

• Prime Number: A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. Examples include 2, 3, 5, 7, 11, and so on.

• Prime Factorization: Prime factorization is the process of expressing a composite number (a number that is not prime) as a product of its prime factors. Every composite number can be uniquely expressed as a product of prime numbers.

Finding the Prime Factors of 225

There are several methods to find the prime factors of 225. Here's a common approach using a factor tree:

1. Start with the smallest prime number: Begin by dividing 225 by the smallest prime number, which is 2. Since 225 is not divisible by 2 (it's an odd number), we move to the next prime number, 3.

2. Divide by 3: 225 divided by 3 is 75. We now have 3 x 75.

3. Continue the process: Now, we need to find the prime factors of 75. 75 is also divisible by 3, resulting in 25. So we have 3 x 3 x 25.

4. Find the prime factors of the remaining number: The remaining number is 25. 25 is not divisible by 3, but it is divisible by 5 (another prime number). 5 x 5 = 25.

5. Final Prime Factorization: Therefore, the prime factorization of 225 is 3 x 3 x 5 x 5, or 3² x 5².

Therefore, the prime factors of 225 are 3 and 5.

Alternative Method: Repeated Division

Another way to achieve the same result is through repeated division:

• Divide 225 by 3: 225 ÷ 3 = 75
• Divide 75 by 3: 75 ÷ 3 = 25
• Divide 25 by 5: 25 ÷ 5 = 5
• The final result is 5, which is a prime number.

This confirms that the prime factorization of 225 is 3 x 3 x 5 x 5 or 3² x 5².

This process demonstrates how to systematically break down a composite number into its fundamental prime components. Understanding this process is vital for various mathematical applications and problem-solving. Remember to always check for divisibility by the smallest prime numbers first, working your way up. This method ensures efficiency and accuracy in finding the prime factors of any number.