75 As A Product Of Prime Factors

75 as a Product of Prime Factors: A Step-by-Step Guide

Finding the prime factorization of a number is a fundamental concept in number theory and is crucial for various mathematical operations. This article will guide you through the process of expressing 75 as a product of its prime factors. Understanding this process will improve your understanding of prime numbers, factorization, and their applications in more advanced mathematical concepts.

What are Prime Factors?

Before we begin, let's quickly define our terms. 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 is the process of expressing a number as a product of its prime factors. This representation is unique for every number (excluding the order of the factors).

Finding the Prime Factors of 75

There are several methods to find the prime factors of 75. Let's explore the most common approach:

Method 1: Factor Tree

The factor tree method is a visual approach that makes it easy to follow the process. We start by finding any two factors of 75. A good starting point is often the smallest prime number, 2. However, 75 is not divisible by 2. The next prime number is 3, and 75 is divisible by 3 (75/3 = 25).

1. Start with 75:

      75
     /  \
    3   25
   

2. Now, we look at the factors 3 and 25. 3 is a prime number, so we leave it as is. 25 is not prime; its factors are 5 and 5.

      75
     /  \
    3   25
       /  \
      5   5
   

3. Both 5s are prime numbers. We have reached the end of our factorization.

Therefore, the prime factorization of 75 is 3 x 5 x 5, or 3 x 5².

Method 2: Division by Prime Numbers

This method involves repeatedly dividing the number by the smallest prime number that divides it evenly until you are left with 1.

1. Start with 75. 75 is not divisible by 2.
2. Divide 75 by 3: 75 / 3 = 25
3. 25 is not divisible by 3.
4. Divide 25 by 5: 25 / 5 = 5
5. Divide 5 by 5: 5 / 5 = 1

We have divided 75 by 3 and then twice by 5. Therefore, the prime factorization of 75 is 3 x 5 x 5, or 3 x 5².

Conclusion: The Prime Factorization of 75

Using both methods, we arrive at the same conclusion: the prime factorization of 75 is 3 x 5². This means 75 can be expressed as the product of its prime factors 3 and 5, with 5 appearing twice. This fundamental concept is essential for understanding various aspects of mathematics, including simplifying fractions, finding greatest common divisors (GCD), and least common multiples (LCM). Understanding this process helps build a strong foundation for more advanced mathematical studies.