Smallest Common Multiple Of 12 And 15 (Under 100)

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Finding the Smallest Common Multiple of 12 and 15 Under 100

Hey guys! Let's dive into a fun math problem today: What's the smallest common multiple of 12 and 15 that's less than 100? This might sound a bit tricky at first, but don't worry, we'll break it down step by step. Understanding multiples and how they work is super useful, not just in math class, but also in everyday life. Think about things like scheduling or dividing tasks – multiples can help you organize things efficiently. So, let’s get started and unlock the mystery of these numbers!

Understanding Multiples

Before we jump into finding the smallest common multiple (SCM), let's quickly recap what multiples are. Simply put, a multiple of a number is what you get when you multiply that number by an integer (a whole number). For example, the multiples of 12 are 12, 24, 36, 48, and so on (12 x 1, 12 x 2, 12 x 3, and so on). Similarly, the multiples of 15 are 15, 30, 45, 60, and so on. Got it? Great! Multiples are like the building blocks of many mathematical concepts, and they're essential for understanding things like fractions, ratios, and, of course, finding common multiples.

Why are Multiples Important?

Understanding multiples is super important in math because they show up everywhere! They help us with:

  • Fractions: When we're adding or subtracting fractions, we need to find a common denominator, which is a multiple of the original denominators.
  • Ratios and Proportions: Multiples help us scale ratios up or down while keeping the relationship between the numbers the same.
  • Scheduling: Think about planning events or organizing tasks. If something happens every 12 days and another thing happens every 15 days, finding the common multiples helps you figure out when they'll happen on the same day.
  • Problem Solving: Many real-world problems, like figuring out how many items to buy to fill containers or dividing things equally, involve multiples.

So, knowing your multiples isn't just about memorizing numbers; it's about having a tool that makes math and life a little easier!

Finding Common Multiples of 12 and 15

Now that we know what multiples are, let's find the common multiples of 12 and 15. Common multiples are simply numbers that are multiples of both 12 and 15. One way to find them is to list out the multiples of each number and see where they overlap. Let's start by listing the first few multiples of 12:

  • 12, 24, 36, 48, 60, 72, 84, 96, 108, ...

And now, let's list the first few multiples of 15:

  • 15, 30, 45, 60, 75, 90, 105, 120, ...

Do you see any numbers that appear in both lists? That's right, 60 is a common multiple of both 12 and 15! But is it the smallest one less than 100? Let's keep going to make sure we find the smallest one.

Identifying Common Multiples

To identify common multiples, we're essentially looking for numbers that both 12 and 15 can divide into evenly. This is like finding a meeting point on a number line where both numbers' multiples intersect. Here’s a more systematic way to do it:

  1. List Multiples: Write out the multiples of each number, like we did before. Go far enough to spot a few overlaps.
  2. Compare Lists: Look at both lists and circle or highlight the numbers that appear in both. These are your common multiples.
  3. Check the Condition: In our case, we want common multiples less than 100. So, we’ll only consider the ones that fit this condition.

We already found 60, but let's see if there are others less than 100. If we continue listing multiples, we’ll find that 120 is also a common multiple (12 x 10 = 120 and 15 x 8 = 120). However, 120 is greater than 100, so it doesn't fit our condition. That means 60 is likely our answer, but let’s be absolutely sure by finding the smallest common multiple.

Finding the Smallest Common Multiple (SCM)

The smallest common multiple (SCM), also known as the least common multiple (LCM), is the smallest number that is a multiple of both numbers. In our lists above, 60 is the first number that appears in both the multiples of 12 and the multiples of 15. So, 60 is a common multiple, but is it the smallest one under 100? To be absolutely sure, let's think about this a bit more.

Methods to Find the SCM

There are a couple of cool ways to find the SCM. One way is by listing multiples, like we did. But there’s another method that’s super helpful, especially for larger numbers: prime factorization.

  1. Prime Factorization: Break down each number into its prime factors. Prime factors are prime numbers that multiply together to give you the original number.
    • For 12, the prime factors are 2 x 2 x 3 (or 2² x 3).
    • For 15, the prime factors are 3 x 5.
  2. Identify Common and Unique Factors: Look for factors that both numbers have in common and factors that are unique to each number. In this case, both 12 and 15 have a 3 as a factor. 12 has two 2s, and 15 has a 5.
  3. Multiply the Highest Powers: To find the SCM, multiply the highest power of each prime factor that appears in either factorization.
    • We have 2² (from 12), 3 (common to both), and 5 (from 15).
    • So, the SCM is 2² x 3 x 5 = 4 x 3 x 5 = 60.

Using this method confirms that 60 is indeed the smallest common multiple of 12 and 15.

The Answer: 60 is the Smallest Common Multiple Under 100

So, after listing multiples and using the prime factorization method, we've confirmed that the smallest common multiple of 12 and 15 that is less than 100 is 60! That means 60 is the smallest number that both 12 and 15 divide into evenly. Cool, right? This kind of problem-solving helps build your math skills and your ability to think logically. You've nailed it!

Wrapping Up and Real-World Connections

Finding the smallest common multiple might seem like a purely math-class kind of thing, but it's actually super useful in lots of everyday situations. Think about these scenarios:

  • Scheduling: Imagine you're organizing a meeting. If one group can meet every 12 days and another every 15 days, knowing the SCM (which is 60) tells you that they'll both be available on the same day every 60 days.
  • Cooking: Some recipes might call for ingredients in fractions, and you need to find a common denominator to combine them correctly. That's where SCM comes in handy!
  • Travel: If you're planning a trip and need to figure out when two buses or trains will arrive at the same station, you might use the concept of common multiples.

So, the next time you're faced with a problem that involves finding common ground or synchronizing events, remember the power of the smallest common multiple. It's more than just a math concept; it's a problem-solving tool that can make your life a little bit easier. Keep practicing, and you'll become a master of multiples in no time!