What Is The Prime Factorization Of 135

What is the Prime Factorization of 135? A Step-by-Step Guide

Finding the prime factorization of a number is a fundamental concept in mathematics, crucial for understanding various topics like greatest common divisors (GCD) and least common multiples (LCM). This article will guide you through the process of determining the prime factorization of 135, explaining each step clearly and concisely. Understanding prime factorization helps build a stronger foundation in number theory and algebra.

What is Prime Factorization?

Prime factorization is the process of expressing a composite number (a number greater than 1 that is not a prime number) as a product of its prime factors. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself. Examples of prime numbers include 2, 3, 5, 7, 11, and so on.

Finding the Prime Factors of 135

To find the prime factorization of 135, we'll use a method of repeated division by prime numbers. Let's start:

1. Start with the smallest prime number, 2: 135 is not divisible by 2 (it's an odd number).

2. Try the next prime number, 3: 135 is divisible by 3 (135 / 3 = 45). So, we can write 135 as 3 x 45.

3. Continue with 45: 45 is also divisible by 3 (45 / 3 = 15). Now we have 135 = 3 x 3 x 15.

4. Factor 15: 15 is divisible by 3 (15 / 3 = 5). This gives us 135 = 3 x 3 x 3 x 5.

5. We've reached a prime number: 5 is a prime number. We can stop here.

Therefore, the prime factorization of 135 is 3 x 3 x 3 x 5, which can also be written as 3³ x 5.

Alternative Method: Factor Tree

You can also use a factor tree to visualize the prime factorization. Here's how it would look for 135:

      135
     /   \
    3    45
       /   \
      3    15
         /   \
        3     5

Following the branches down to the prime numbers at the bottom (3, 3, 3, and 5), we arrive at the same prime factorization: 3³ x 5.

Conclusion

The prime factorization of 135 is 3³ x 5. This method can be applied to find the prime factorization of any composite number. Understanding this process is essential for tackling more complex mathematical problems involving number theory and algebra. Remember to always divide by prime numbers, starting from the smallest, until you're left with only prime numbers. This ensures you find the complete and accurate prime factorization.