Numbers Divisible By 5 Between 74 And 25556 - Math Problem Solved!

Hey there, math enthusiasts! Ever found yourself pondering how many numbers within a certain range are divisible by a specific number? It's a classic mathematical puzzle, and today, we're going to crack it. We'll tackle the question: How many numbers lie between 74 and 25,556 that are perfectly divisible by 5? Get ready to dive into the world of numbers, where we'll explore the step-by-step method to solve this problem, ensuring you not only understand the solution but also the underlying concepts. So, grab your thinking caps, and let's get started on this numerical adventure!

Understanding Divisibility and Arithmetic Progressions

Before we jump into solving the problem directly, let's build a solid foundation. Understanding the concept of divisibility is key. A number is divisible by another if the remainder after division is zero. For example, 10 is divisible by 5 because 10 ÷ 5 = 2 with no remainder. This forms the basis of our quest to find numbers divisible by 5. Next, we need to grasp the idea of arithmetic progressions. An arithmetic progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant. Think of it like a staircase, where each step is the same height. In our case, the numbers divisible by 5 form an AP (e.g., 5, 10, 15, 20...). Recognizing this pattern is crucial because it allows us to use formulas and techniques specific to APs, making our task much easier. These core concepts—divisibility and arithmetic progressions—are the cornerstones of our approach to solving this problem. Understanding them thoroughly will not only help you solve this particular question but also equip you with valuable tools for tackling other mathematical challenges. It's like having the right key to unlock a treasure chest of mathematical knowledge!

Pinpointing the First and Last Numbers Divisible by 5

Okay, guys, let's get down to business! The first step in our numerical treasure hunt is to identify the starting and ending points within our range (74 to 25,556) that are divisible by 5. Think of it as marking the boundaries of our search area. To find the first number greater than 74 that's divisible by 5, we can simply round up 74 to the nearest multiple of 5. In this case, that's 75 (75 ÷ 5 = 15). This is our starting point, the first stepping stone in our sequence. Now, let's find the last number less than or equal to 25,556 that's divisible by 5. Similarly, we round down 25,556 to the nearest multiple of 5. A quick calculation tells us that 25,555 is the magic number (25,555 ÷ 5 = 5,111). This marks the end of our sequence, the final piece of the puzzle within our range. These two numbers, 75 and 25,555, are crucial because they define the boundaries of our arithmetic progression. We now know where our sequence starts and where it ends, setting the stage for the next step in our calculation. It's like having the first and last pieces of a jigsaw puzzle – we're well on our way to completing the picture!

Applying the Arithmetic Progression Formula

Now that we've identified the first (75) and last (25,555) numbers in our sequence, it's time to put our arithmetic progression (AP) knowledge to work. Remember, an AP is a sequence where the difference between consecutive terms is constant – in our case, 5. To find the total number of terms (numbers divisible by 5) in this AP, we'll use a nifty formula: n = (last term - first term) / common difference + 1. This formula might look a bit intimidating at first, but it's actually quite straightforward. Let's break it down: * Last term: 25,555 (the largest number in our range divisible by 5) * First term: 75 (the smallest number in our range divisible by 5) * Common difference: 5 (the constant difference between consecutive multiples of 5) Now, let's plug these values into the formula: n = (25,555 - 75) / 5 + 1. Performing the calculation, we get: n = 25,480 / 5 + 1 = 5,096 + 1 = 5,097. So, there you have it! The formula reveals that there are 5,097 numbers between 74 and 25,556 that are divisible by 5. This formula is a powerful tool in our mathematical arsenal, allowing us to efficiently count the terms in any arithmetic progression. By understanding and applying this formula, we've successfully navigated a key step in solving our problem. It's like having a secret code that unlocks the answer!

Conclusion: The Grand Total of Numbers Divisible by 5

Alright, folks, we've reached the finish line! After our numerical journey, meticulously identifying the first and last terms, and expertly applying the arithmetic progression formula, we've arrived at our answer. Drumroll, please... There are a grand total of 5,097 numbers between 74 and 25,556 that are perfectly divisible by 5. This result, 5,097, corresponds to option B in the original question. Congratulations to everyone who followed along and understood the process! But more importantly, let's reflect on the broader significance of what we've accomplished. This exercise wasn't just about finding a single answer; it was about honing our problem-solving skills and deepening our understanding of mathematical concepts. We've explored the principles of divisibility, grasped the essence of arithmetic progressions, and mastered a formula that can be applied to a wide range of similar problems. These skills are invaluable, not just in mathematics, but in any field that requires logical thinking and analytical reasoning. So, the next time you encounter a numerical puzzle, remember the steps we've taken today – you've got the tools to conquer it! Keep exploring, keep questioning, and keep those mathematical gears turning!

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Numbers Divisible by 5 Between 74 and 25556 - Math Problem Solved!

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