What Are The Prime Factors Of 75

What Are the Prime Factors of 75? A Step-by-Step Guide

Finding the prime factors of a number is a fundamental concept in mathematics, crucial for understanding number theory and simplifying complex calculations. This article will guide you through the process of determining the prime factors of 75, explaining the method and underlying principles clearly. Understanding this process will empower you to tackle similar problems with ease.

What are Prime Factors?

Before we dive into finding the prime factors of 75, let's define the key 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. Every composite number (a number that is not prime) can be uniquely expressed as a product of prime numbers.

Finding the Prime Factors of 75

There are several ways to find the prime factors of 75. Here's a common and straightforward method:

1. Start with the smallest prime number: Begin by dividing 75 by the smallest prime number, which is 2. Since 75 is not divisible by 2 (it's an odd number), we move to the next smallest prime number.

2. Divide by 3: 75 is divisible by 3 (75 ÷ 3 = 25). Therefore, 3 is one of the prime factors.

3. Continue the process: Now we have 25. 25 is not divisible by 3, but it is divisible by the next prime number, 5 (25 ÷ 5 = 5). So, 5 is another prime factor.

4. The final prime factor: We are left with 5, which is itself a prime number.

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

Alternative Method: Factor Tree

Another helpful visual method is using a factor tree. This method is especially useful for larger numbers.

      75
     /  \
    3   25
       /  \
      5    5

The factor tree shows the breakdown of 75 into its prime factors: 3, 5, and 5.

Conclusion: Prime Factors of 75

In summary, the prime factors of 75 are 3 and 5. The prime factorization is expressed as 3 x 5². Understanding how to find prime factors is a crucial skill in various mathematical applications, from simplifying fractions to solving algebraic equations. By mastering this technique, you'll enhance your mathematical problem-solving abilities significantly. Remember to always start with the smallest prime number and continue dividing until you're left with only prime numbers.