Find The Prime Factorization Of 75

Finding the Prime Factorization of 75: A Step-by-Step Guide

Finding the prime factorization of a number is a fundamental concept in mathematics, crucial for understanding various mathematical operations and concepts. This article will guide you through the process of finding the prime factorization of 75, explaining the steps clearly and concisely. We'll explore different methods and ensure you understand the underlying principles. Understanding prime factorization is vital for simplifying fractions, finding the greatest common divisor (GCD), and the least common multiple (LCM) of numbers.

What is Prime Factorization?

Prime factorization is the process of expressing a composite number (a number greater than 1 that is not prime) 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.

Method 1: Using a Factor Tree

The factor tree method is a visual approach to finding prime factorization. It involves repeatedly breaking down a number into smaller factors until all the factors are prime numbers.

1. Start with the number 75: Write 75 at the top of your factor tree.

2. Find two factors of 75: 75 can be factored into 3 and 25. Write 3 and 25 as branches below 75.

3. Continue factoring: 3 is a prime number, so we circle it. 25 is not prime; it can be factored into 5 and 5. Write 5 and 5 as branches below 25.

4. All factors are prime: Both 5s are prime numbers, so we circle them.

Your factor tree should look something like this:

      75
     /  \
    3   25
       /  \
      5    5

5. Write the prime factorization: The prime factorization of 75 is 3 x 5 x 5, or 3 x 5².

Method 2: Repeated Division

This method involves repeatedly dividing the number by its smallest prime factor until the result is 1.

1. Start with the number 75: Divide 75 by its smallest prime factor, which is 3. 75 / 3 = 25.

2. Continue dividing: The result is 25. The smallest prime factor of 25 is 5. 25 / 5 = 5.

3. Final division: The result is 5, which is a prime number. 5 / 5 = 1.

4. Write the prime factorization: The prime factors are 3 and 5 (twice), so the prime factorization of 75 is 3 x 5 x 5 or 3 x 5².

Conclusion:

Both methods demonstrate that the prime factorization of 75 is 3 x 5². Understanding prime factorization is a valuable skill in mathematics, providing a foundation for more advanced concepts. Remember to practice both methods to solidify your understanding and become proficient in finding the prime factorization of any number. This skill is essential for various mathematical applications, including simplifying fractions and solving equations.