Finding Common Ground: The Common Factor of 56 and 72

Have you ever found yourself knee-deep in numbers, scratching your head over the concept of common factors? It might feel like solving a puzzle—but don’t worry, we’re here to make this simple! Today, we’re focusing on two numbers that often come up in studies and exams: 56 and 72. So, what’s the common factor of these two? Let’s break it down!

To start, a common factor is a number that divides two or more numbers without leaving a remainder. In our case, we want to find the greatest common factor (GCF) of 56 and 72. You might think, "Surely both numbers have many factors!" And that’s true! But what we want is the largest one, the crème de la crème of factors. So let’s see how we can find it.

Step One: Factor Each Number
To discover commonalities, we first need to break each number down to its prime factors.

• For 56, you can break it down like this:
• 56 = 2 × 28
• Then, factor 28:
• 28 = 2 × 14
• Continuing, we get:
• 14 = 2 × 7
• So, ultimately, the prime factorization of 56 is:
• 56 = 2³ × 7

Now onto 72:

• 72 can be broken down as follows:
• 72 = 8 × 9 (and since 8 and 9 can be factored further...)
• 8 = 2³
• 9 = 3²
• Therefore, the prime factorization of 72 is:
• 72 = 2³ × 3²

Step Two: List the Factors
Now that we’ve factored both numbers, let’s list out the factors:

• From 56’s factors, we have: 1, 2, 4, 7, 8, 14, 28, 56
• From 72’s factors, we have: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Step Three: Identify Common Factors
Looking at both lists, the factors they share are: 1, 2, 4, and 8. Here's a little fun fact for you: of these common factors, 8 is the largest. It’s like finding the most valuable treasure in a chest filled with coins!

Now, let’s pause for a moment—does this suddenly feel easier? It’s a bit like navigating through a maze, isn’t it? You sometimes have to take a few turns before finding the right path!

The Answer is... 8!
To answer the question, the greatest common factor of 56 and 72 is indeed 8. While 24 could also divide these two numbers, it’s not the largest common factor, which is what we’re after. Isn’t math fascinating? You might think you’re just calculating numbers, but you’re actually unraveling deeper connections between them.

And what about those options we mentioned earlier?

• Option A, 24? Nice try, but it doesn’t quite make the cut.
• Option C, 3? Not a factor of either number.
• And Option D, 6? It’s simply overshadowed by our winning number—8.

So there you have it! Not only have we identified the common factor here, but we’ve also armed you with some handy skills for your future math challenges.

As you prepare for your College Algebra CLEP exam, remember this little exercise. There’s so much more to explore: other algebraic concepts, different types of equations, and finding those hidden connections. Keep practicing! And don't forget—it’s okay to find things challenging; the learning journey is just as important as the destination!

Now, go on and tackle those numbers with confidence! You’ve got this!