FPB 48 Dan 60: Cara Mudah Menemukannya!

Okay, guys, let's dive into finding the faktor persekutuan terkecil (FPB), which is basically the greatest common factor (GCF), of 48 and 60. This is a super useful skill, especially when you're trying to simplify fractions or solve problems involving division. So, stick around, and we'll break it down step by step!

Understanding Faktor Persekutuan Terkecil (FPB)

Before we jump into the nitty-gritty, let's make sure we're all on the same page about what FPB actually means. The Faktor Persekutuan Terkecil (FPB), or Greatest Common Factor (GCF), is the largest number that can divide evenly into two or more numbers. Think of it like this: you're looking for the biggest piece you can cut two different-sized cakes into so that you don't have any crumbs left over.

Why is this important? Well, FPB is super handy in many areas of math. For example, when you're simplifying fractions, finding the FPB of the numerator and denominator allows you to reduce the fraction to its simplest form. Imagine you have the fraction 48/60. By finding the FPB, you can divide both numbers and make the fraction much easier to work with. Also, FPB helps in solving various algebraic problems, especially those involving factorization and simplification of expressions. Understanding FPB lays a strong foundation for more advanced mathematical concepts.

There are a couple of ways to find the FPB. We can list the factors of each number and find the largest one they have in common. Or, we can use prime factorization, which is breaking down each number into its prime factors and then identifying the common ones. We'll go through both methods so you can choose the one that clicks best with you. The goal is to find the largest number that perfectly divides both 48 and 60, making problem-solving easier and more efficient. So, let's get started and explore how to find this magical number!

Method 1: Listing Factors

Alright, let's kick things off with the first method: listing the factors. This approach is pretty straightforward and easy to grasp, especially if you're just starting out with FPB. What we're going to do is list all the numbers that divide evenly into both 48 and 60. Then, we'll simply pick out the biggest one they share. Simple, right?

First, let's list the factors of 48. A factor is a number that divides exactly into 48 without leaving any remainder. So, we start with 1 (because 1 divides into everything), then 2, 3, 4, 6, 8, 12, 16, 24, and finally 48 itself. That's quite a few factors, but don't worry, we'll keep things organized. Now, let's do the same for 60. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Okay, we've got both lists ready.

Next, we need to compare these two lists and find the factors they have in common. Looking at both lists, we can see that 1, 2, 3, 4, 6, and 12 appear in both. These are the common factors of 48 and 60. But remember, we're not just looking for any common factor; we want the greatest one. So, among these common factors (1, 2, 3, 4, 6, and 12), the largest one is 12. Therefore, the FPB of 48 and 60 is 12. See, not too hard, right? This method is great for smaller numbers because it's easy to keep track of all the factors. However, it can get a bit cumbersome when you're dealing with larger numbers that have many factors. But for now, we've nailed it using the listing factors method!

Method 2: Prime Factorization

Now, let's move on to the second method: prime factorization. This method might sound a bit more technical, but trust me, it's super useful, especially when you're dealing with larger numbers. Prime factorization involves breaking down each number into its prime factors. Remember, a prime number is a number that has only two factors: 1 and itself (e.g., 2, 3, 5, 7, etc.).

So, let's start by breaking down 48 into its prime factors. We can do this by dividing 48 by the smallest prime number that divides it evenly, which is 2. So, 48 ÷ 2 = 24. Then, we divide 24 by 2 again: 24 ÷ 2 = 12. Keep going: 12 ÷ 2 = 6, and 6 ÷ 2 = 3. Finally, 3 is a prime number, so we stop there. So, the prime factorization of 48 is 2 × 2 × 2 × 2 × 3, or 2⁴ × 3.

Now, let's do the same for 60. We start by dividing 60 by 2: 60 ÷ 2 = 30. Then, 30 ÷ 2 = 15. Next, 15 isn't divisible by 2, so we move on to the next prime number, which is 3: 15 ÷ 3 = 5. And 5 is a prime number, so we stop. Thus, the prime factorization of 60 is 2 × 2 × 3 × 5, or 2² × 3 × 5.

Now comes the fun part: identifying the common prime factors. Looking at the prime factorizations of 48 (2⁴ × 3) and 60 (2² × 3 × 5), we can see that they both have 2s and 3s as prime factors. To find the FPB, we take the lowest power of each common prime factor. So, for 2, we take 2² (since it's the lowest power between 2⁴ and 2²), and for 3, we take 3 (since it's the same power in both factorizations). Then, we multiply these together: 2² × 3 = 4 × 3 = 12. So, the FPB of 48 and 60 is 12. This method is especially handy when dealing with larger numbers because it breaks down the problem into smaller, more manageable parts.

Therefore...

So, after walking through both methods – listing factors and prime factorization – we've arrived at the same answer: the faktor persekutuan terkecil (FPB) of 48 and 60 is 12. Whether you prefer the simplicity of listing factors or the systematic approach of prime factorization, you now have two solid ways to tackle FPB problems. Keep practicing, and you'll become a pro in no time! And that's all there is to it, guys! Now you know how to find the FPB of 48 and 60. Go forth and conquer those math problems!