Greatest Common Factor Of 72 And 90

Finding the Greatest Common Factor (GCF) of 72 and 90

Finding the greatest common factor (GCF), also known as the greatest common divisor (GCD), of two numbers is a fundamental concept in mathematics with applications ranging from simplifying fractions to solving algebraic equations. This article will guide you through several methods to determine the GCF of 72 and 90, explaining each step clearly so you can easily apply these techniques to other number pairs. Understanding GCFs is crucial for various mathematical operations, and mastering this skill will significantly enhance your mathematical abilities.

What is the Greatest Common Factor?

The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into all of them without leaving a remainder. In simpler terms, it's the biggest number that's a factor of both 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 GCF of 12 and 18 is 6.

Methods for Finding the GCF of 72 and 90

We'll explore three common methods to find the GCF of 72 and 90:

1. Listing Factors Method

This method involves listing all the factors of each number and identifying the largest factor they share.

• Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
• Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

By comparing the lists, we can see that the common factors are 1, 2, 3, 6, 9, and 18. The largest of these common factors is 18. Therefore, the GCF of 72 and 90 is 18. This method is straightforward for smaller numbers but can become cumbersome with larger numbers.

2. Prime Factorization Method

This method involves finding the prime factorization of each number and then multiplying the common prime factors raised to their lowest powers.

• Prime factorization of 72: 2³ × 3² (2 x 2 x 2 x 3 x 3)
• Prime factorization of 90: 2 × 3² × 5 (2 x 3 x 3 x 5)

The common prime factors are 2 and 3. The lowest power of 2 is 2¹ and the lowest power of 3 is 3². Therefore, the GCF is 2¹ × 3² = 2 × 9 = 18. This method is more efficient for larger numbers than listing factors.

3. Euclidean Algorithm

The Euclidean algorithm is a highly efficient method for finding the GCF, particularly useful for larger numbers. It's based on repeatedly applying the division algorithm.

1. Divide the larger number (90) by the smaller number (72): 90 ÷ 72 = 1 with a remainder of 18.
2. Replace the larger number with the smaller number (72) and the smaller number with the remainder (18).
3. Repeat the process: 72 ÷ 18 = 4 with a remainder of 0.
4. Since the remainder is 0, the GCF is the last non-zero remainder, which is 18.

The Euclidean algorithm provides a systematic and efficient way to find the GCF, even for very large numbers.

Conclusion

We have demonstrated three different methods to determine the greatest common factor of 72 and 90, all leading to the same result: 18. Choosing the most appropriate method depends on the size of the numbers and your familiarity with each technique. Understanding the concept of GCF and mastering these methods is essential for various mathematical applications and problem-solving. Remember to practice regularly to build proficiency and confidence in your mathematical skills.