Mastering the GCF: A Friendly Guide to Florida Teacher Certification Examinations

What is the greatest common factor (GCF) of the numbers 36, 135, and 144?

• A. 3
• B. 6
• C. 9
• D. 12

Answer: (C)

To determine the greatest common factor (GCF) of the numbers 36, 135, and 144, it is essential to analyze the prime factorization of each number first. Starting with 36, its prime factorization is: - 36 = 2 × 2 × 3 × 3 = \(2^2 \times 3^2\) Next, for 135, the prime factorization is: - 135 = 3 × 3 × 3 × 5 = \(3^3 \times 5^1\) Lastly, for 144, the prime factorization is: - 144 = 2 × 2 × 2 × 3 × 3 = \(2^3 \times 3^2\) Now, to find the GCF, we identify the common prime factors in all three factorizations and take the lowest power of each common factor: - The prime factor 2 appears in 36 and 144 but not in 135, so it is not included in the GCF. - The prime factor 3 appears in all three numbers. The lowest power of 3 in the factorizations is \(3^2\) (from

When it comes to getting ready for the Florida Teacher Certification Examinations (FTCE), understanding concepts like the greatest common factor (GCF) can be a step towards success. So, what’s the big deal about the GCF? Picture this: You're gearing up for a test, and math questions pop up. You want to tackle them with confidence, right? That's where mastering the GCF comes into play. Let’s take a closer look at how to find the GCF of numbers like 36, 135, and 144, shall we?

First things first, let's break down our numbers using prime factorization—a fancy term, but trust me, it’s not as scary as it sounds! For instance, our friend 36 can be broken down into its prime factors:

• 36 = 2 × 2 × 3 × 3 = (2^2 \times 3^2)

Next up, we’ve got 135:

• 135 = 3 × 3 × 3 × 5 = (3^3 \times 5^1)

And then there’s 144 to factor:

• 144 = 2 × 2 × 2 × 3 × 3 = (2^3 \times 3^2)

Alright, so we’ve unlocked the prime factorizations—didn’t that feel good? Now, here’s what we do next: we need to identify which prime factors are common across all three numbers. Here’s the kicker:

• The prime factor 2 pops up in 36 and 144, but it plays no role in 135. So, we can just toss it out of our GCF equation (after all, no one likes an outsider at a party).

• Now, let’s shine the spotlight on 3. It appears in all three numbers, and the lowest power of 3 among our factorizations is (3^2).

So, drumroll please… the GCF of 36, 135, and 144 is 9. Yup, that’s right—good ol’ number 9!

Now, imagine you’re sitting in front of your students, and they’re staring blankly at the board while you’re breaking down the GCF concept. It’s one thing to solve the problem yourself, but how do you convey that understanding to others? Here’s a thought: dive into real-world examples. Use relatable scenarios—like what happens when you divide a pizza and want to ensure everyone gets an equal share. Everyone loves pizza; it’s a win-win!

Just think about it for a second. You know the satisfaction of cracking a challenging math problem? That’s the kind of confidence you want your students to feel when faced with similar questions on their own tests, isn’t it? By making the concepts engaging and relevant, you're not just preparing them for the FTCE; you're also setting them up for a lifetime of problem-solving skills.

So, as you start prepping for the Florida Teacher Certification Examinations, remember that each tiny detail—like finding the GCF—builds a greater foundation for your teaching journey. Keep at it, and before you know it, you and your students will become GCF pros together. Now, get out there and let that math magic shine!