What Is Prime Factorization Of 75

What is the Prime Factorization of 75? A Step-by-Step Guide

Finding the prime factorization of a number is a fundamental concept in mathematics, particularly useful in simplifying fractions, finding greatest common divisors (GCD), and least common multiples (LCM). This article will guide you through the process of finding the prime factorization of 75, explaining the method and concepts involved. Understanding this will equip you with a valuable skill for various mathematical problems.

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 natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 2, 3, 5, 7, 11 are prime numbers. The prime factorization of a number is unique; it's like a number's unique fingerprint.

Finding the Prime Factors of 75

There are several ways to find the prime factorization of a number. The most common method is using a factor tree. Let's apply this to 75:

1. Find the smallest prime factor: The smallest prime number is 2. However, 75 is not divisible by 2 (it's an odd number). The next prime number is 3. 75 is divisible by 3 (75 / 3 = 25).

2. Branch out: We can represent this as a branch in our factor tree:

       75
      /  \
     3   25
   

3. Continue factoring: Now we need to factor 25. 25 is not divisible by 3, but it is divisible by 5 (25 / 5 = 5).

4. Complete the tree: Our factor tree now looks like this:

       75
      /  \
     3   25
        /  \
       5    5
   

5. Identify the prime factors: We've reached the point where all the branches end in prime numbers (3 and 5). Therefore, the prime factorization of 75 is 3 x 5 x 5, or 3 x 5².

Therefore, the prime factorization of 75 is 3 x 5²

Alternative Method: Repeated Division

Another method involves repeatedly dividing the number by its prime factors until you reach 1.

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

This also shows that the prime factorization of 75 is 3 x 5 x 5 or 3 x 5².

Applications of Prime Factorization

Understanding prime factorization is crucial for various mathematical operations:

• Simplifying Fractions: Finding the GCD of the numerator and denominator allows you to simplify fractions to their lowest terms.
• Finding LCM: The LCM is essential for adding or subtracting fractions with different denominators.
• Number Theory: Prime factorization is a cornerstone of number theory, used in cryptography and other advanced mathematical fields.

By mastering prime factorization, you enhance your mathematical problem-solving skills and gain a deeper understanding of number properties. Now you can confidently tackle the prime factorization of other numbers using the factor tree or repeated division method!