Greatest Common Factor Of 24 And 90

Finding the Greatest Common Factor (GCF) of 24 and 90: A Comprehensive Guide

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest number that divides exactly into two or more numbers without leaving a remainder. Finding the GCF is a fundamental concept in mathematics with applications spanning various fields, from simplifying fractions to solving algebraic equations. This article will explore multiple methods for determining the GCF of 24 and 90, providing a deep understanding of the process and its underlying principles.

Understanding the Concept of GCF

Before diving into the methods, let's solidify our understanding of what the GCF represents. Imagine you have 24 apples and 90 oranges. You want to divide both fruits into identical groups, with each group containing the same number of apples and the same number of oranges. The largest possible size of these groups is determined by the GCF of 24 and 90. This means we're looking for the biggest number that can perfectly divide both 24 and 90.

Method 1: Prime Factorization

This is arguably the most fundamental method for finding the GCF. It involves breaking down each number into its prime factors – prime numbers that multiply together to give the original number. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself (e.g., 2, 3, 5, 7, 11...).

Steps:

1. Find the prime factorization of 24:

   24 = 2 x 2 x 2 x 3 = 2³ x 3

2. Find the prime factorization of 90:

   90 = 2 x 3 x 3 x 5 = 2 x 3² x 5

3. Identify common prime factors: Both 24 and 90 share a '2' and a '3'.

4. Calculate the GCF: Multiply the common prime factors raised to their lowest power. In this case, we have one '2' and one '3'.

   GCF(24, 90) = 2¹ x 3¹ = 6

Therefore, the greatest common factor of 24 and 90 is 6. This means we can divide both 24 apples and 90 oranges into 6 identical groups, each containing 4 apples and 15 oranges.

Advantages and Disadvantages of Prime Factorization

Advantages:

• Fundamental understanding: This method provides a clear understanding of the prime composition of the numbers.
• Applicable to larger numbers: It works effectively even with larger numbers, although the factorization process might become more time-consuming.
• Basis for other methods: The principles of prime factorization are foundational to other GCF-finding techniques.

Disadvantages:

• Can be tedious for large numbers: Finding the prime factors of very large numbers can be a lengthy process.
• Requires knowledge of prime numbers: A basic understanding of prime numbers is essential for successful application.

Method 2: Listing Factors

This method involves listing all the factors (divisors) of each number and then identifying the largest common factor. A factor is a number that divides another number without leaving a remainder.

Steps:

1. List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

2. List the factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

3. Identify common factors: The common factors of 24 and 90 are 1, 2, 3, and 6.

4. Determine the GCF: The largest common factor is 6.

Therefore, the GCF(24, 90) = 6.

Advantages and Disadvantages of Listing Factors

Advantages:

• Simple and intuitive: This method is easy to understand and requires minimal mathematical background.
• Suitable for smaller numbers: It works well for smaller numbers where listing factors is manageable.

Disadvantages:

• Inefficient for larger numbers: Listing all factors becomes increasingly time-consuming and impractical for larger numbers.
• Prone to errors: Manually listing factors increases the risk of missing some factors, leading to an incorrect GCF.

Method 3: Euclidean Algorithm

The Euclidean algorithm is an efficient method for finding the GCF of two numbers, particularly useful for larger numbers. It's based on the principle that the GCF of two numbers doesn't change if the larger number is replaced by its difference with the smaller number. This process is repeated until the two numbers are equal, and that number is the GCF.

Steps:

1. Divide the larger number (90) by the smaller number (24):

   90 ÷ 24 = 3 with a remainder of 18

2. Replace the larger number with the remainder (18): Now we find the GCF of 24 and 18.

3. Repeat the process:

   24 ÷ 18 = 1 with a remainder of 6

4. Repeat again:

   18 ÷ 6 = 3 with a remainder of 0

5. The GCF is the last non-zero remainder: The last non-zero remainder is 6.

Therefore, the GCF(24, 90) = 6.

Advantages and Disadvantages of the Euclidean Algorithm

Advantages:

• Efficient for larger numbers: This method is significantly more efficient than listing factors or prime factorization for larger numbers.
• Systematic and reliable: The algorithm is systematic and less prone to errors.

Disadvantages:

• Less intuitive: The underlying principle might not be as immediately clear as the other methods.
• Requires division skills: A good understanding of division and remainders is necessary.

Applications of Finding the GCF

The ability to find the greatest common factor has various practical applications in mathematics and beyond:

• Simplifying fractions: The GCF is crucial for simplifying fractions to their lowest terms. For example, the fraction 24/90 can be simplified to 4/15 by dividing both the numerator and denominator by their GCF, which is 6.

• Solving algebraic equations: The GCF plays a role in factoring algebraic expressions and solving equations.

• Word problems: Many word problems involving division and grouping require finding the GCF to solve them efficiently. (Recall our apple and orange example.)

• Geometry: The GCF can be used in geometrical problems involving the dimensions of shapes and finding the largest possible square tiles to cover a rectangular area.

• Number theory: The GCF is a fundamental concept in number theory, used in various advanced mathematical theorems and proofs.

Conclusion

Finding the greatest common factor of two numbers, like 24 and 90, is a fundamental skill with wide-ranging applications. While multiple methods exist, the choice of method often depends on the size of the numbers and the level of mathematical understanding. Prime factorization offers a deep insight into number composition, while the Euclidean algorithm provides an efficient approach for larger numbers. Understanding these methods empowers you to tackle various mathematical problems and appreciate the elegance and practicality of the GCF concept. Mastering this skill forms a strong foundation for more advanced mathematical concepts and problem-solving. By applying these methods, you can confidently determine the greatest common factor and apply this knowledge to a variety of mathematical and real-world scenarios. Remember to choose the method best suited to the task at hand – for smaller numbers, listing factors might suffice, while for larger numbers, the Euclidean algorithm offers a more efficient solution.