Numbers Divisible By 3 And 7 Between 1 And 5000 Solution

Introduction

In this article, we will explore a fascinating mathematical problem that involves determining the count of numbers between 1 and 5000 that are divisible by both 3 and 7. This exercise delves into the realms of number theory, specifically focusing on divisibility rules and the concept of the least common multiple (LCM). Understanding these principles is crucial not only for solving this particular problem but also for tackling a wide range of mathematical challenges. We will break down the problem step-by-step, providing a clear and concise solution while also highlighting the underlying mathematical concepts. Whether you're a student looking to enhance your problem-solving skills or simply a math enthusiast curious about number theory, this article will offer valuable insights and a comprehensive approach to solving this type of problem. By the end of this exploration, you will have a solid grasp of how to identify numbers divisible by multiple factors and how to apply this knowledge to various mathematical scenarios. Join us on this journey as we unravel the intricacies of divisibility and number theory, making the complex world of mathematics more accessible and engaging. This problem not only tests our ability to perform calculations but also our understanding of fundamental mathematical principles. So, let's dive in and discover the elegant solution to this intriguing question.

Understanding Divisibility Rules

Before we dive into the specific problem at hand, it's crucial to understand the concept of divisibility rules. A number is said to be divisible by another number if the remainder is zero after division. For instance, 12 is divisible by 3 because 12 ÷ 3 = 4, with no remainder. Similarly, 21 is divisible by 7 because 21 ÷ 7 = 3, again with no remainder. However, when we consider numbers divisible by both 3 and 7, we need to think about a key concept: the least common multiple (LCM). The LCM of two numbers is the smallest number that is a multiple of both. In this case, we need to find the LCM of 3 and 7. Since 3 and 7 are both prime numbers, their LCM is simply their product, which is 3 × 7 = 21. This means that any number divisible by both 3 and 7 must also be divisible by 21. Now that we have established this crucial point, we can reframe our original problem. Instead of searching for numbers divisible by 3 and 7 separately, we can focus on finding numbers between 1 and 5000 that are divisible by 21. This significantly simplifies our task. Understanding the LCM allows us to consolidate two divisibility requirements into a single, more manageable condition. This is a common technique in number theory and can be applied to various problems involving multiple divisibility criteria. By grasping this fundamental concept, we can approach the problem with a clearer strategy and a more efficient method of calculation. In the next section, we will apply this understanding to determine the number of integers between 1 and 5000 that are divisible by 21, ultimately leading us to the solution.

Determining the Range and Approach

Now that we know we need to find numbers divisible by 21 within the range of 1 to 5000, let's outline our approach. The core idea is to determine how many multiples of 21 fall within this range. We can start by finding the largest multiple of 21 that is less than or equal to 5000. To do this, we can divide 5000 by 21 and consider the quotient (the whole number result of the division). When we divide 5000 by 21, we get approximately 238.095. This tells us that 21 goes into 5000 about 238 times. However, we are only interested in the whole number part, which is 238. This means that 238 × 21 is the largest multiple of 21 that is less than or equal to 5000. To verify this, we can multiply 238 by 21, which gives us 4998. This confirms that 4998 is indeed the largest multiple of 21 within our range. Now, the question becomes: how many multiples of 21 are there between 1 and 4998? Since we start with 21 × 1 = 21 and go up to 21 × 238 = 4998, it's clear that there are 238 multiples of 21 in this range. Each multiple corresponds to an integer from 1 to 238. Therefore, the number of integers between 1 and 5000 that are divisible by both 3 and 7 (or equivalently, by 21) is 238. This straightforward approach, using division and the concept of multiples, allows us to efficiently solve the problem. In the next section, we will present the final answer and summarize the steps we took to arrive at the solution.

Calculation and Solution

To recap, we've established that a number divisible by both 3 and 7 must be divisible by their least common multiple, which is 21. We then determined that we need to find the number of multiples of 21 between 1 and 5000. We divided 5000 by 21 and obtained approximately 238.095. Taking the integer part of this result, we get 238. This means that 238 × 21 is the largest multiple of 21 that is less than or equal to 5000. Multiplying 238 by 21, we get 4998, which confirms our calculation. Now, we know that there are 238 multiples of 21 between 1 and 5000. These multiples are 21 × 1, 21 × 2, 21 × 3, and so on, up to 21 × 238. Each of these multiples is divisible by both 3 and 7. Therefore, the answer to the question is 238. This concludes our calculation and provides a definitive solution to the problem. The process we followed demonstrates a systematic approach to solving divisibility problems. By understanding the underlying mathematical principles, such as the least common multiple, and applying basic arithmetic operations, we can efficiently find the answer. In the final section, we will summarize our findings and highlight the key takeaways from this problem-solving exercise. This will help solidify the concepts we've covered and provide a comprehensive understanding of the solution.

Final Answer and Conclusion

In summary, we set out to find the number of integers between 1 and 5000 that are divisible by both 3 and 7. Through our step-by-step approach, we first recognized that a number divisible by both 3 and 7 must be divisible by their least common multiple (LCM), which is 21. We then focused on finding the number of multiples of 21 within the given range. By dividing 5000 by 21, we obtained approximately 238.095. Taking the integer part, we found that there are 238 multiples of 21 between 1 and 5000. Therefore, the final answer to the question is (b) 238. This problem illustrates the importance of understanding fundamental mathematical concepts, such as divisibility rules and the least common multiple. By applying these principles, we were able to efficiently solve a seemingly complex problem. The process we followed can be generalized to solve other similar problems involving divisibility and number theory. The key takeaways from this exercise include the importance of identifying the least common multiple when dealing with multiple divisibility criteria, and the utility of division in determining the number of multiples within a given range. We hope this article has provided a clear and comprehensive understanding of the solution to this problem, and that it has enhanced your problem-solving skills in the realm of mathematics. Remember, practice and a solid understanding of basic concepts are crucial for tackling mathematical challenges with confidence. This exploration into number theory underscores the elegance and practicality of mathematical principles in solving real-world problems.