Smallest 3-Digit Number Divisible By 15, 25, 35 And Traffic Light Synchronization

Introduction

In this article, we will delve into two interesting mathematical problems. First, we'll explore how to determine the smallest 3-digit number that is exactly divisible by 15, 25, and 35. This involves understanding the concept of the Least Common Multiple (LCM) and its application in finding numbers divisible by multiple factors. Second, we'll tackle a real-world scenario involving traffic lights and their synchronization. We'll investigate how to calculate when traffic lights at different intersections will change simultaneously, again using the principles of LCM. These problems highlight the practical applications of mathematical concepts in everyday situations. Understanding these concepts can help in various fields, from basic arithmetic to more complex problem-solving in engineering and logistics.

Determining the Smallest 3-Digit Number Divisible by 15, 25, and 35

To find the smallest 3-digit number exactly divisible by 15, 25, and 35, we need to employ the concept of the Least Common Multiple (LCM). The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set. In simpler terms, it's the smallest number that all the given numbers can divide into without leaving a remainder. Finding the LCM is a fundamental skill in number theory and has practical applications in various areas such as scheduling, resource allocation, and even music theory.

Finding the Least Common Multiple (LCM)

The first step is to determine the LCM of 15, 25, and 35. We can do this using the prime factorization method. Prime factorization involves breaking down each number into its prime factors. A prime factor is a prime number that divides the given number exactly. For instance, the prime factors of 15 are 3 and 5, since 15 = 3 × 5. Similarly, we find the prime factors of 25 and 35.

• 15 = 3 × 5
• 25 = 5 × 5 = 5²
• 35 = 5 × 7

Once we have the prime factorization of each number, we identify the highest power of each prime factor that appears in any of the factorizations. In this case, the prime factors are 3, 5, and 7. The highest power of 3 is 3¹, the highest power of 5 is 5², and the highest power of 7 is 7¹. The LCM is then the product of these highest powers.

LCM (15, 25, 35) = 3¹ × 5² × 7¹ = 3 × 25 × 7 = 525

Thus, the LCM of 15, 25, and 35 is 525. This means that 525 is the smallest number that is divisible by all three numbers. However, we are looking for the smallest 3-digit number. So, we need to find the smallest multiple of the LCM that is a 3-digit number.

Finding the Smallest 3-Digit Multiple

Since the LCM, 525, is a 3-digit number, it seems like we have our answer. However, 525 is not the smallest 3-digit number divisible by 15, 25, and 35. It's simply the LCM. To find the smallest 3-digit multiple, we need to consider multiples of the LCM. The smallest 3-digit number is 100. We need to find the smallest multiple of 525 that is greater than or equal to 100. Since 525 itself is a 3-digit number, we made an error in our understanding of the problem. The correct approach is to find the LCM first (which we have done as 525) and then determine the smallest 3-digit number that is divisible by 15, 25, and 35.

Our initial LCM calculation is correct: LCM(15, 25, 35) = 525. The issue is that 525 is already a 3-digit number, and we need to find the smallest 3-digit number divisible by these numbers. We should have realized that we need to find a smaller number. The correct approach is to find the LCM as we did (525), then find the smallest 3-digit number divisible by 15, 25, and 35. This requires us to find the smallest 3-digit number that when divided by 15, 25, and 35 leaves no remainder.

To correct our approach, we recognize that any number divisible by 15, 25, and 35 must be a multiple of their LCM. We've already calculated the LCM as 525. Now, we need to find the smallest 3-digit number that is a factor of 15, 25, and 35. This is where the initial understanding of the problem was flawed. We need to find the smallest 3-digit number that is a multiple of a number divisible by 15, 25, and 35, not necessarily a multiple of their LCM directly.

Let's reconsider the prime factors: 15 = 3 x 5, 25 = 5 x 5, and 35 = 5 x 7. We still correctly found the LCM as 525. The problem is in the interpretation of what we need to find. We need a 3-digit number divisible by 15, 25, and 35. Since 525 is the LCM, any multiple of 525 will also be divisible by 15, 25, and 35. However, 525 is too large. We need the smallest 3-digit number.

To find the correct answer, we should divide the smallest 3-digit number (100) by the LCM (525) and find the next whole number. 100 divided by 525 is approximately 0.19. The next whole number is 1. So, 1 x 525 = 525. This is a 3-digit number, but it's not the smallest. We made another mistake in our logic.

The key to solving this is understanding that the number we're looking for must be a multiple of the LCM. We found the LCM to be 525. Now, instead of dividing 100 by 525, we should consider the smallest 3-digit number (100) and find the smallest multiple of a common factor of 15, 25, and 35 that results in a 3-digit number. The common factor is 5. Multiples of 5 will not directly lead us to the answer since we need a number divisible by 15, 25, and 35. We need a different approach.

We have correctly calculated the LCM as 525. To find the smallest 3-digit number divisible by 15, 25, and 35, we need to find the smallest 3-digit number that is a multiple of 525. However, 525 itself is already a 3-digit number, and it's divisible by 15, 25, and 35. Therefore, we need to check if there's a smaller 3-digit number that also meets the criteria.

Let's go back to the basics. We need a number that is divisible by 15, 25, and 35. This means it must be divisible by the prime factors of these numbers: 3, 5, 5, 5, and 7. The LCM, 525, ensures this. However, we need the smallest 3-digit number. It seems we keep circling back to 525, but let's try a different approach.

If we consider the factors, we need a number divisible by 3, 5² (25), and 7. The LCM approach ensures this. Since 525 is the LCM, any smaller number divisible by all three (15, 25, 35) is impossible. The confusion arises from trying to find a smaller number than the LCM that still meets the criteria. The LCM is, by definition, the smallest number divisible by all given numbers.

Therefore, the smallest 3-digit number divisible by 15, 25, and 35 is indeed 525.

Traffic Light Synchronization

Now, let's shift our focus to the second problem: traffic light synchronization. This is a practical application of LCM in urban planning and traffic management. The problem states that traffic lights at three different points of intersection change after every 40 seconds, 48 seconds, and 72 seconds, respectively. If they change simultaneously at a particular time, we want to find out when they will change simultaneously again. This involves finding the LCM of the time intervals at which the lights change.

Calculating the LCM for Traffic Light Intervals

The intervals at which the traffic lights change are 40 seconds, 48 seconds, and 72 seconds. To find when they will change simultaneously again, we need to determine the LCM of these three numbers. This will give us the time interval after which all three lights will change at the same time.

We again use the prime factorization method:

• 40 = 2 × 2 × 2 × 5 = 2³ × 5
• 48 = 2 × 2 × 2 × 2 × 3 = 2⁴ × 3
• 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²

Now, we identify the highest power of each prime factor:

• Highest power of 2: 2⁴
• Highest power of 3: 3²
• Highest power of 5: 5¹

LCM (40, 48, 72) = 2⁴ × 3² × 5¹ = 16 × 9 × 5 = 720

Therefore, the LCM of 40, 48, and 72 is 720 seconds. This means that the traffic lights will change simultaneously every 720 seconds.

Converting Seconds to Minutes

To make the answer more understandable, we can convert 720 seconds into minutes. There are 60 seconds in a minute, so we divide 720 by 60:

720 seconds / 60 seconds/minute = 12 minutes

So, the traffic lights will change simultaneously every 12 minutes. If they changed simultaneously at a certain time, they will change simultaneously again after 12 minutes.

Conclusion

In this article, we've tackled two problems that demonstrate the practical application of the Least Common Multiple (LCM). First, we successfully determined that the smallest 3-digit number divisible by 15, 25, and 35 is 525. This required a solid understanding of prime factorization and the concept of LCM. We navigated through some initial misinterpretations to arrive at the correct solution, highlighting the importance of careful problem analysis. Second, we calculated that traffic lights at three different intersections, changing at intervals of 40, 48, and 72 seconds, will change simultaneously every 12 minutes. This showcased how LCM is used in real-world scenarios like traffic management.

These examples emphasize the versatility of mathematical concepts and their relevance in everyday life. Understanding LCM not only helps in solving mathematical problems but also provides valuable insights into various practical situations. Whether it's finding the smallest number divisible by a set of factors or synchronizing events that occur at different intervals, the principles of LCM offer a powerful tool for problem-solving.