Problem 23 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of numbers. $$ 40,32,24 $$

Step-by-Step Solution

Answer

The GCF of 40, 32, and 24 is 8.

Step 1: Find the Prime Factorization

First, we need to find the prime factorization of each number in the list. - **40** can be factored as: \[40 = 2^3 imes 5\] - **32** can be factored as: \[32 = 2^5\] - **24** can be factored as: \[24 = 2^3 imes 3\]

Step 2: Identify Common Prime Factors

Identify the prime factors common to all numbers. Here, we can see that 2 is the only common prime factor.- **40**: \(2^3\)- **32**: \(2^5\)- **24**: \(2^3\)

Step 3: Choose the Smallest Exponent

For each common prime factor, choose the smallest exponent found in the factorizations of the numbers. For the prime factor 2, the exponents are 3, 5, and 3. Therefore, we choose 3.

Step 4: Calculate the GCF

Now use the smallest exponent to calculate the GCF. Multiply the factor by itself raised to the smallest exponent.\[\text{GCF} = 2^3 = 8\]

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime number components. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, in the exercise we were given the numbers 40, 32, and 24, each of which was broken down into prime numbers:

• 40 as \(2^3 \times 5\)
• 32 as \(2^5\)
• 24 as \(2^3 \times 3\)

The purpose of prime factorization, particularly when finding the Greatest Common Factor (GCF), is to reduce complex numbers into simple, manageable pieces. Once we have the prime factors, it makes identifying common factors and calculating the GCF much easier. By consistently practicing prime factorization, you'll improve your ability to manipulate and understand numbers in their simplest form.

Common Prime Factors

In the prime factorization of each number, common prime factors are the ones that appear in all given numbers. After conducting the prime factorization of 40, 32, and 24, the only common prime factor identified is 2. This means that 2 is the smallest building block shared by all these numbers. Understanding which prime factors are shared among numbers is essential in determining the GCF, as only these common factors contribute to it.
When identifying common prime factors, it’s crucial to line up the factors for easy comparison. In this case:

• 40: \(2^3\)
• 32: \(2^5\)
• 24: \(2^3\)

All three have the factor 2 raised to some power. Remember, common prime factors streamline calculations by narrowing the focus to only the factors essential for finding the GCF.

Exponent

An exponent refers to the number of times a number, known as the base, is multiplied by itself. It is a compact notation used in mathematics. In the context of our exercise, the exponent is crucial for determining which common prime factors contribute to the Greatest Common Factor (GCF).
In our example, the common prime factor among 40, 32, and 24 is 2. The exponents for the factor 2 are:

• 40: \(2^3\)
• 32: \(2^5\)
• 24: \(2^3\)

To find the GCF, we select the smallest exponent for our common prime factor. Here, we choose 3 because it's the lowest exponent for 2 across all numbers. Therefore, \(2^3\) or 8 is our GCF.
Using the smallest exponent ensures that the GCF is the highest number that can evenly divide each of the original numbers. Exponents help simplify calculations, particularly when handling larger numbers or more complex factorizations. Understanding exponents will greatly enhance your mathematical fluency and effectiveness.