Finding The Least Common Multiple Of 168 And 1116 Divisible By 210

Hey there, math enthusiasts! Today, we're diving into a fun problem: finding the smallest number that is both a multiple of 168 and 1116, and also divisible by 210. Sounds like a mouthful, right? But don't worry, we'll break it down step by step. This is a classic problem involving the Least Common Multiple (LCM) and divisibility, which are fundamental concepts in number theory. Understanding these concepts is crucial not only for solving mathematical problems but also for various real-world applications, such as scheduling, resource allocation, and even music theory. So, let's put on our thinking caps and get started!

Understanding the Problem

Before we jump into the calculations, let's make sure we understand what the problem is asking. We need to find a number that satisfies three conditions:

1. It must be a multiple of 168.
2. It must be a multiple of 1116.
3. It must be divisible by 210.

Essentially, we're looking for a common multiple of 168 and 1116, but not just any common multiple – the smallest one. And, this smallest common multiple has an extra requirement: it has to be perfectly divisible by 210. This adds an interesting twist to the typical LCM problem. To tackle this, we'll first find the LCM of 168 and 1116, and then we'll make sure that the result is divisible by 210. If it's not, we'll need to find the smallest multiple of the LCM that is divisible by 210. This might sound a bit complicated, but trust me, it's manageable once we break it down into smaller steps. We're essentially combining two concepts here: finding the LCM and checking for divisibility. So, let's get started with the first step: finding the LCM of 168 and 1116.

Step 1: Finding the LCM of 168 and 1116

Okay, so the first thing we need to do is figure out the Least Common Multiple (LCM) of 168 and 1116. There are a couple of ways we can do this, but one of the most reliable methods is using prime factorization.

First, let's break down 168 and 1116 into their prime factors:

• 168 = 2 x 2 x 2 x 3 x 7 = 2³ x 3 x 7
• 1116 = 2 x 2 x 3 x 93 = 2² x 3 x 3 x 31 = 2² x 3² x 31

Now, to find the LCM, we need to take the highest power of each prime factor that appears in either factorization and multiply them together. Let's see what we've got:

• The highest power of 2 is 2³ (from 168).
• The highest power of 3 is 3² (from 1116).
• We also have 7 (from 168) and 31 (from 1116).

So, the LCM of 168 and 1116 is 2³ x 3² x 7 x 31 = 8 x 9 x 7 x 31. Now, let's multiply those numbers together: 8 x 9 = 72, 72 x 7 = 504, and finally, 504 x 31 = 15624. Therefore, the LCM of 168 and 1116 is 15624. This means that 15624 is the smallest number that is a multiple of both 168 and 1116. But we're not done yet! Remember, we have one more condition to satisfy: the number must also be divisible by 210. So, let's move on to the next step and check if 15624 fits the bill.

Step 2: Checking Divisibility by 210

Alright, we've found the LCM of 168 and 1116, which is 15624. Now, the next crucial step is to check if this LCM is divisible by 210. To do this, we simply divide 15624 by 210 and see if we get a whole number (i.e., no remainder). Grab your calculators, guys, let's do this!

15624 ÷ 210 = 74.4

Hmm, we got 74.4, which is not a whole number. This means that 15624 is not divisible by 210. Bummer! But don't worry, this is just a small bump in the road. We know that the number we're looking for needs to be a multiple of the LCM (15624) and also divisible by 210. So, what do we do? We need to find the smallest multiple of 15624 that is divisible by 210. Think of it like this: we're going up the multiples of 15624 until we find one that plays nicely with 210.

To figure out the smallest multiple, we can look at the prime factors of 210 and compare them to the prime factors of 15624. This will help us identify what's "missing" in 15624 to make it divisible by 210. Let's find the prime factorization of 210:

210 = 2 x 3 x 5 x 7

Now, let's recall the prime factorization of 15624:

15624 = 2³ x 3² x 7 x 31

Comparing these, we see that 15624 has 2³, 3², 7, and 31 as prime factors, while 210 has 2, 3, 5, and 7. The key difference is that 15624 doesn't have a factor of 5. That's our missing piece! So, to make 15624 divisible by 210, we need to multiply it by 5. This is because multiplying by 5 will introduce the missing prime factor, ensuring divisibility by 210. Let's do that in the next step.

Step 3: Finding the Multiple Divisible by 210

Okay, we've identified that the LCM of 168 and 1116 (which is 15624) isn't divisible by 210 because it's missing a prime factor of 5. So, as we discussed, we need to multiply 15624 by 5 to make it divisible by 210. Let's do the math:

15624 x 5 = 78120

So, 78120 is a multiple of 15624, and since we multiplied by 5, it should now be divisible by 210. But let's double-check just to be sure. We don't want to jump to conclusions, right?

78120 ÷ 210 = 372

Woohoo! We got a whole number (372), which means 78120 is indeed divisible by 210. Now, let's recap what we've done. We found the LCM of 168 and 1116, which was 15624. Then, we realized it wasn't divisible by 210, so we figured out that we needed to multiply it by 5. This gave us 78120, which we confirmed is divisible by 210. Therefore, 78120 is a common multiple of 168 and 1116, and it's also divisible by 210. But is it the smallest such number? Well, since we multiplied the least common multiple by the smallest factor needed to achieve divisibility by 210, we can confidently say that 78120 is indeed the smallest number that satisfies all the conditions. We're almost there! Now, let's put it all together and state our final answer.

Final Answer

Alright, guys, we've made it to the end! We've successfully navigated through the problem, step by step, and now we have our final answer. To recap, we were looking for the smallest number that is a multiple of both 168 and 1116, and also divisible by 210. We found the LCM of 168 and 1116, which was 15624. Then, we realized that 15624 wasn't divisible by 210, so we multiplied it by 5 to get 78120. We confirmed that 78120 is indeed divisible by 210, and since we used the LCM and the smallest necessary factor, we know that this is the smallest number that meets all the criteria.

Therefore, the smallest multiple of 168 and 1116 that is divisible by 210 is 78120.

Awesome job, everyone! We tackled a challenging problem involving LCM and divisibility, and we came out victorious. Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. Understanding the concepts of prime factorization, LCM, and divisibility is crucial, and practicing these skills will make you a math whiz in no time. Keep up the great work, and I'll see you in the next math adventure!