Gcf Of 75 And 90

Finding the Greatest Common Factor (GCF) of 75 and 90: A Comprehensive Guide

Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics. It's a skill crucial for simplifying fractions, solving algebraic equations, and understanding number theory. This comprehensive guide will explore various methods to determine the GCF of 75 and 90, explaining each step clearly and providing a deep understanding of the underlying principles. We will also explore the broader implications of GCFs and answer frequently asked questions.

Understanding Greatest Common Factor (GCF)

Before diving into the calculation, let's define what a greatest common factor actually is. The GCF of two or more numbers is the largest number that divides evenly into all of them without leaving a remainder. Think of it as the biggest common "building block" of those numbers. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 18 are 1, 2, 3, 6, 9, and 18. The common factors of 12 and 18 are 1, 2, 3, and 6. The greatest of these common factors is 6, therefore the GCF of 12 and 18 is 6.

Method 1: Listing Factors

The most straightforward method, especially for smaller numbers like 75 and 90, is to list all the factors of each number and identify the largest common one.

Factors of 75: 1, 3, 5, 15, 25, 75

Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Comparing the two lists, we can see that the common factors are 1, 3, 5, and 15. The largest of these is 15.

Therefore, the GCF of 75 and 90 is 15.

Method 2: Prime Factorization

This method is more efficient for larger numbers and provides a deeper understanding of the numbers' structure. It involves breaking down each number into its prime factors – numbers divisible only by 1 and themselves.

Prime Factorization of 75:

75 = 3 x 25 = 3 x 5 x 5 = 3 x 5²

Prime Factorization of 90:

90 = 2 x 45 = 2 x 3 x 15 = 2 x 3 x 3 x 5 = 2 x 3² x 5

Now, identify the common prime factors and their lowest powers:

Both 75 and 90 contain a factor of 3 (with the lowest power being 3¹)
Both 75 and 90 contain a factor of 5 (with the lowest power being 5¹)

To find the GCF, multiply these common prime factors with their lowest powers:

GCF(75, 90) = 3¹ x 5¹ = 3 x 5 = 15

Method 3: Euclidean Algorithm

The Euclidean Algorithm is a highly efficient method for finding the GCF, especially for larger numbers. It's based on repeated division.

Divide the larger number (90) by the smaller number (75):

90 ÷ 75 = 1 with a remainder of 15
Replace the larger number with the smaller number (75) and the smaller number with the remainder (15):
Repeat the division:

75 ÷ 15 = 5 with a remainder of 0

Since the remainder is 0, the GCF is the last non-zero remainder, which is 15.

Illustrative Examples and Applications of GCF

Understanding GCF has practical applications beyond simple mathematical exercises. Let's consider some examples:

Simplifying Fractions: To simplify the fraction 75/90, we find the GCF of 75 and 90 (which is 15). Dividing both the numerator and the denominator by 15 gives us the simplified fraction 5/6.
Dividing Objects Evenly: Imagine you have 75 apples and 90 oranges. You want to divide them into identical bags, with each bag containing the same number of apples and oranges. The GCF (15) tells you that you can create 15 bags, each containing 5 apples and 6 oranges.
Tile Arrangements: Suppose you're tiling a room with square tiles. The room measures 75 inches by 90 inches. You want to use the largest possible square tiles without cutting any. The GCF (15) indicates that the largest square tile you can use is 15 inches by 15 inches.
Algebraic Simplification: In algebra, finding the GCF is essential for simplifying expressions. For example, to factor the expression 75x + 90y, we find the GCF of 75 and 90 (15), resulting in 15(5x + 6y).

Beyond Two Numbers: Finding the GCF of Multiple Numbers

The methods described above can be extended to find the GCF of more than two numbers. Using prime factorization is generally the most efficient approach.

Let's find the GCF of 75, 90, and 105:

Prime Factorization of 75: 3 x 5²
Prime Factorization of 90: 2 x 3² x 5
Prime Factorization of 105: 3 x 5 x 7

The common prime factors are 3 and 5. The lowest power of 3 is 3¹, and the lowest power of 5 is 5¹. Therefore, the GCF(75, 90, 105) = 3 x 5 = 15.

Frequently Asked Questions (FAQ)

Q: What if the GCF of two numbers is 1?

A: If the GCF of two numbers is 1, the numbers are said to be relatively prime or coprime. This means they share no common factors other than 1.

Q: Is there a shortcut for finding the GCF of very large numbers?

A: For very large numbers, advanced algorithms like the Euclidean Algorithm or specialized software are recommended. Manually applying prime factorization can become very time-consuming.

Q: Why is the GCF important in mathematics?

A: The GCF is a fundamental concept used in various areas of mathematics, including simplifying fractions, solving equations, and understanding number theory. Its applications extend to other fields like computer science and cryptography.

Q: Can the GCF of two numbers be one of the numbers?

A: Yes, this happens when one number is a multiple of the other. For example, the GCF of 15 and 30 is 15.

Conclusion

Finding the greatest common factor is a valuable skill with widespread applications. Whether you use the listing factors method, prime factorization, or the Euclidean Algorithm, understanding the process will help you navigate various mathematical problems efficiently and effectively. Remember, the choice of method depends largely on the size and complexity of the numbers involved. This comprehensive guide has equipped you with the knowledge and tools necessary to confidently tackle GCF calculations and appreciate their significance in mathematics. With practice, you'll find these calculations become increasingly intuitive and straightforward.