Problem 24 - Elementary and Intermediate Algebra

Question

Find the GCF of each list of numbers. $$ 28,35,21 $$

Step-by-Step Solution

Answer

The GCF is 7.

Step 1: Understand the Problem

We need to find the Greatest Common Factor (GCF) of the numbers 28, 35, and 21. The GCF is the largest number that divides all the given numbers exactly.

Step 2: Prime Factorization

To find the GCF, we'll first perform prime factorization on each number:- **28**: \(28 = 2^2 \times 7\)- **35**: \(35 = 5 \times 7\)- **21**: \(21 = 3 \times 7\)

Step 3: Identify Common Factors

From the prime factorizations, identify the common prime factors among the numbers: - The common factor is **7** (since it's the only prime factor common to all three numbers).

Step 4: Conclude the Solution

The GCF is the product of all common prime factors identified. Since the only common factor is 7, the GCF is 7.

Key Concepts

Prime Factorization

When finding the Greatest Common Factor (GCF), one helpful method is **prime factorization**. This technique involves breaking down each given number into its simplest building blocks—its prime numbers.
Prime numbers are those greater than 1 that can only be divided by 1 and themselves. Examples include 2, 3, 5, and 7. Think of them as the "atoms" of the number world; they can't be broken down any further into other whole numbers.

Here's a simple way to do prime factorization:

• Divide the number by the smallest prime (e.g., 2, if it's even).
• Keep dividing by the same prime number until it no longer divides evenly.
• Move on to the next smallest prime number.
• Repeat until the result is a prime number.

In our example, the prime factorization of 28 is \(2^2 \times 7\), for 35 it's \(5 \times 7\), and for 21 it's \(3 \times 7\). Remember, it's like breaking down a cake into slices, and each slice should be as small as possible!🔬

Common Factors

Once you've got your numbers broken down through prime factorization, the next step is 1dentifying common factors**. These are factors that appear in the prime factorizations of all numbers in your list. They are the "shared slices" that each number can "agree on."

For any set of numbers, simply compare the lists of prime factors you've found:

• Look for primes that appear in all lists.
• The commonality in these lists shows the factors each number shares.

In our example, the common factor is the number 7. Why? Because 7 appears in each list of prime factors—it's the only slice shared between all numbers. For some students, visualizing this similarity might help, as it’s akin to finding matching socks in a drawer. 🧦 The common factors provide the essential parts needed to calculate the GCF.

Divisibility

Divisibility is an important concept when talking about the GCF. A number is **divisible** by another if you can divide them without getting a remainder.

To test for divisibility:

• Check if the division of one number by another results in a whole number.
• If yes, it's divisible; if not, there's a remainder.

For instance, let's check divisibility by looking at our numbers 28, 35, and 21. After finding the common factor, you might test it by dividing each number:

• 28 divided by 7 equals 4.
• 35 divided by 7 equals 5.
• 21 divided by 7 equals 3.

All these divisions result in whole numbers, demonstrating that each original number is divisible by 7 without leaving a remainder. 🌟 Understanding divisibility helps reinforce why 7 is the greatest common factor—it's the largest number that can "go into" each of the numbers exactly, with nothing left behind.