Write The Prime Factorization Of 75

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

This article will guide you through the process of finding the prime factorization of 75. Understanding prime factorization is crucial in various mathematical concepts, including simplifying fractions, finding least common multiples (LCM), and greatest common divisors (GCD). We'll break down the process simply and clearly.

What is Prime Factorization?

Prime factorization is the process of expressing a 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. The prime factorization of a number is unique, meaning there's only one way to express it as a product of primes (ignoring the order of the factors).

Finding the Prime Factors of 75

There are several methods to find the prime factorization of a number. One common method is to use a factor tree. Let's use this method for 75:

1. Start with the smallest prime number: The smallest prime number is 2. However, 75 is not divisible by 2 (it's not an even number). The next prime number is 3. Is 75 divisible by 3? Yes! 75 divided by 3 is 25.

2. Continue factoring: Now we have 3 x 25. 3 is a prime number, so we leave it as it is. However, 25 is not a prime number. The next prime number to consider is 5. Is 25 divisible by 5? Yes! 25 divided by 5 is 5.

3. Prime factors found: We now have 3 x 5 x 5. Since 3 and 5 are both prime numbers, we have found the prime factors of 75.

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

Alternative Method: Repeated Division

Another method is repeated division. We start by dividing the number by the smallest prime number that divides it evenly and continue until we are left with 1.

• 75 ÷ 3 = 25
• 25 ÷ 5 = 5
• 5 ÷ 5 = 1

This method also leads us to the same prime factorization: 3 x 5 x 5 or 3 x 5².

Conclusion:

The prime factorization of 75 is 3 x 5². Mastering prime factorization is a valuable skill that simplifies many mathematical operations and deepens your understanding of number theory. Remember to practice using different methods to solidify your understanding. Try finding the prime factorization of other numbers using the techniques outlined above. You can apply this knowledge to solve problems involving LCM, GCD, and fraction simplification. Understanding prime numbers and their factorization will provide a strong foundation for your mathematical journey.