Understanding the Prime Factors of 144

What is the sum of the prime factors of 144?

• A. 12
• B. 14
• C. 16
• D. 18

Answer: (B)

To find the sum of the prime factors of 144, we first need to determine the prime factorization of the number. Start by breaking down 144: - 144 is an even number, so we divide it by 2. This gives us 72. - Continuing to factor 72, we again see it is even, giving 36 when divided by 2. - Dividing 36 by 2 yields 18. - Dividing 18 by 2 gives us 9. - Now, 9 can be factored further, as it is 3 times 3. Hence, the prime factorization of 144 is: \[ 144 = 2^4 \times 3^2 \] From this factorization, we identify the unique prime factors, which are 2 and 3. Next, we find the sum of these prime factors: \[ 2 + 3 = 5 \] This indicates we've identified the prime factors correctly, but it seems we need the sum considering their multiplicity rather than uniqueness. When calculating with multiplicity: \[ (2 + 2 + 2 + 2 + 3 + 3) = 4 \times 2

When it comes to tackling the Kaplan Nursing Entrance Exam, a grasp of math concepts like prime factors can truly be beneficial. So, let’s pull back the curtain on how to break down the number 144 and uncover its prime components—starting right now.

First up, you might be asking, “What exactly is prime factorization?” Simply put, it’s the process of expressing a number as the product of its prime numbers. Think of prime numbers as the building blocks of whole numbers—like the essential ingredients in a recipe!

Now, let’s get back to our number, 144. It’s like that hefty textbook you had to carry around in nursing school; it’s a big deal. We’ll start breaking it down by dividing it by 2—since 144 is even. And voilà, we get 72. But wait, there’s more!

As we continue factoring, we note that 72 is also even, so 72 divided by 2 gives us 36. Keep running with this; dividing 36 by 2 gives us 18, and dividing 18 by 2 takes us to 9. With 9, we change course because it’s not even. 9 equals 3 times 3—so there it is!

Putting this all together, we discover that the prime factorization of 144 is:

[ 144 = 2^4 \times 3^2 ]

Now, let’s take a moment to reflect. From this factorization, we identify our unique prime factors. Can you guess what they are? Yep, you got it! They are 2 and 3. This is a critical point that many overlook when learning.

Next step: calculating the sum of these prime factors. That’s super straightforward, right? Just add them up:

[ 2 + 3 = 5 ]

But hold your horses! There's a twist in the plot here; we're not just looking for that simple sum. When we're factoring, we often need to consider their multiplicity, which means counting each occurrence. So here’s the thought process:

[ (2 + 2 + 2 + 2 + 3 + 3) = 4 \times 2 + 2 \times 3 = 8 + 6 = 14 ]

Aha! So the sum, considering how many times each prime factor appears, is 14. That's the gold nugget of information you need for tackling related problems on your exam.

Remember, engaging with prime factors isn't just a numerical exercise; it's a gateway to honing your math skills. By breaking down a seemingly complicated concept into manageable pieces, you not only empower yourself with knowledge but also make the study process enjoyable. So, as you prep for the Kaplan Nursing Entrance Exam, take this foundation and build on it. With practice, you can conquer mathematical challenges, one prime factor at a time!