Mastering Remainders: Smallest & Largest 3-Digit Numbers

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Mastering Remainders: Smallest & Largest 3-Digit Numbers

Hey there, math enthusiasts and curious minds! Ever stumbled upon a brain-teaser that sounds simple but makes you scratch your head for a bit? Well, today, we're diving deep into a super cool mathematical puzzle that’s all about three-digit numbers and their remainders when divided by different values. This isn't just about crunching numbers; it's about understanding the underlying logic, the elegant patterns that emerge, and how seemingly complex problems can be broken down into manageable, even fun, steps. We're on a quest to pinpoint the smallest and largest three-digit natural numbers that share a peculiar trait: they all give the same non-zero remainder when divided by 8, 12, and 21. Sounds like a mouthful, right? But trust me, guys, by the time we're done, you'll feel like a true number wizard!

This kind of problem is fantastic because it touches on fundamental concepts in number theory – things like divisibility, multiples, and that super-important friend, the Least Common Multiple (LCM). These aren't just abstract ideas found in dusty textbooks; they're the building blocks for everything from cryptography to scheduling algorithms. So, understanding them in a practical, problem-solving context like this one really solidifies your grasp. Imagine you're trying to figure out a repeating pattern in a series of events, or perhaps you're an engineer needing to synchronize different components – the principles we're about to explore are at the heart of those real-world applications. We'll walk through this journey together, making sure every concept is clear, every step is justified, and every "aha!" moment is celebrated. We'll break down the initial statement, clarify what each condition means, and then systematically build up our solution, piece by elegant piece. Get ready to flex those mental muscles and discover the beauty hidden within numbers!

Cracking the Code: Understanding the Problem

Alright, let's kick things off by really understanding the problem statement. This is crucial, guys, because misinterpreting even one tiny detail can send us down the wrong rabbit hole. The core of our challenge is to find specific natural numbers. What does "natural number" mean in this context? Simply put, these are the counting numbers: 1, 2, 3, and so on, extending infinitely. No negatives, no fractions, just good old positive integers. Next, the problem specifies three-digit numbers. This immediately sets our working range. We're not looking at single-digit wonders or massive numbers in the thousands; we're confined to the numbers from 100 up to 999. This constraint is incredibly important, as it helps us narrow down our search considerably. Think of it like setting the boundaries on a treasure hunt – you know exactly where to start digging!

Now, for the really interesting part: these numbers, when divided by 8, 12, and 21, must all yield the same remainder. And here's the kicker: this remainder must be different from zero. This "non-zero" condition is key. If the remainder were zero, it would simply mean the number is perfectly divisible by 8, 12, and 21. But because it's a non-zero remainder, it adds an extra layer of complexity and elegance to our solution. Let's briefly recap what a remainder actually is. When you divide one number (the dividend) by another (the divisor), the result is a quotient and sometimes a leftover, which is the remainder. For example, 10 divided by 3 is 3 with a remainder of 1. That remainder r always has to be less than the divisor. This property will be absolutely vital when we determine the possible values for our common remainder. So, in essence, we're searching for numbers between 100 and 999 that, no matter if you divide them by 8, 12, or 21, always leave behind the exact same little leftover, and that leftover can't be nothing! It's like a secret handshake among numbers, and we're about to uncover it. This intricate requirement for a consistent, non-zero remainder is what makes this a truly engaging number theory problem. We're not just looking for any number; we're looking for numbers that fit a very specific, multi-faceted pattern. Understanding these initial conditions perfectly is our launchpad for success.

The Magic of LCM: Unlocking the Core Concept

Alright, with the problem clearly defined, it’s time to introduce our secret weapon: the Least Common Multiple, or LCM. If you're a bit rusty on LCM, no worries – we're going to break it down. When a number N gives the same remainder r when divided by different numbers (let's say a, b, and c), it means something super important. It means that if you subtract that remainder r from N, the resulting number, N - r, must be perfectly divisible by a, b, and c. Think about it: if N = ak + r, then N - r = ak, which is clearly a multiple of a. The same applies for b and c. So, N - r is a common multiple of 8, 12, and 21. And to make our lives easier, we want to find the smallest such common multiple, which is, you guessed it, the Least Common Multiple. The LCM is like the foundational building block for all other common multiples. Any number that is a common multiple of 8, 12, and 21 must be a multiple of their LCM. This is a powerful concept in divisibility rules and is absolutely central to solving problems like ours. Without finding the LCM, we'd be testing numbers aimlessly – but with it, we get a clear, structured path forward.

Now, how do we calculate the LCM for our specific divisors: 8, 12, and 21? The most reliable method, especially for numbers with multiple factors, is prime factorization. This involves breaking each number down into its prime components.

  • First, let's take 8. Its prime factors are 2 * 2 * 2, which we write as 2^3.
  • Next, 12. Its prime factors are 2 * 2 * 3, or 2^2 * 3.
  • Finally, 21. Its prime factors are 3 * 7.

To find the LCM, we take all unique prime factors from these numbers (which are 2, 3, and 7) and raise each to its highest power observed in any of the factorizations.

  • For the prime factor 2, the highest power we see is 2^3 (from 8).
  • For the prime factor 3, the highest power we see is 3^1 (from 12 and 21).
  • For the prime factor 7, the highest power we see is 7^1 (from 21).

So, the LCM is 2^3 * 3^1 * 7^1. Let's multiply these out: 8 * 3 * 7 = 24 * 7 = 168.

Boom! 168 is our magic number. This means N - r must be a multiple of 168. So, we can express N - r as 168m, where m is any positive integer. Consequently, our number N can be written in the form N = 168m + r. This general form is the backbone of our solution. It bundles all the divisibility requirements neatly into one equation. This shows the true power of the Least Common Multiple in simplifying complex number theory problems, allowing us to move from individual division statements to a single, elegant mathematical expression that encapsulates all conditions. Truly, a game-changer for our quest to find those special three-digit numbers!

Step-by-Step LCM Calculation

Just to make sure we're all on the same page, let's quickly recap the prime factorization for the LCM of 8, 12, and 21:

  1. Factorize 8: 8 is 2 x 2 x 2 = 2^3
  2. Factorize 12: 12 is 2 x 2 x 3 = 2^2 x 3^1
  3. Factorize 21: 21 is 3 x 7 = 3^1 x 7^1

To find the LCM, we pick the highest power for each unique prime factor present:

  • Highest power of 2: 2^3
  • Highest power of 3: 3^1
  • Highest power of 7: 7^1

Multiply them together: LCM = 2^3 * 3^1 * 7^1 = 8 * 3 * 7 = 168. Easy peasy!

Finding the Remainder's Limits: What r Can Be

Alright, guys, we’ve got our general form for N: N = 168m + r. Now, before we start plugging in values, we need to talk about that little r – our remainder. Remember, the problem clearly states that the numbers, when divided by 8, 12, and 21, all give the same non-zero remainder. This "non-zero" part is crucial, meaning r cannot be 0. But what are the upper limits for r? This is where the remainder property comes into play. When you divide a number N by a divisor D, the remainder r must always be less than D. If r were equal to or greater than D, it would mean you could divide at least one more time, and r wouldn't be the true remainder.

In our problem, the number N is divided by 8, 12, and 21. Since N gives the same remainder r for all three divisions, r must satisfy the remainder property for each of these divisors.

  • When N is divided by 8, the remainder r must be less than 8. So, r < 8.
  • When N is divided by 12, the remainder r must be less than 12. So, r < 12.
  • When N is divided by 21, the remainder r must be less than 21. So, r < 21.

For r to satisfy all three conditions simultaneously, it must be less than the smallest of the three divisors. In our case, the smallest divisor is 8. Therefore, r must be less than 8. Combining this with the "non-zero" condition, we know that r can be any integer from 1 to 7, inclusive. That is, r ∈ {1, 2, 3, 4, 5, 6, 7}. This is a vital step because it defines the exact range of possible remainders, giving us a finite set of values to consider. Without this clear understanding of the remainder's limits, our search for N would be much broader and less efficient. This constraint is one of the elegant little tricks in number theory that allows us to narrow down possibilities quickly. It’s a testament to the precision required in mathematical problem-solving and highlights why understanding basic definitions is never boring – it’s powerful! So now we know not just the form of our numbers (168m + r), but also the exact values r can take. We're getting closer and closer to unveiling those special three-digit numbers!

Assembling the Puzzle: The General Form of Our Numbers

Okay, we've pieced together some pretty vital clues, and now it's time to bring them all together into a powerful mathematical equation that will guide us to our answers. We know from our deep dive into the Least Common Multiple that any number N that leaves the same remainder r when divided by 8, 12, and 21 implies that N - r is a multiple of their LCM. We meticulously calculated the LCM of 8, 12, and 21 to be 168. This means N - r must be a multiple of 168. We can write this elegantly as:

N - r = 168 * m

Here, m is an integer. What kind of integer, you ask? Since N is a natural number (positive), and r is also positive (our non-zero remainder, smallest being 1), N - r must also be positive. Therefore, m must be a positive integer. So, m can be 1, 2, 3, and so on. If m were 0, N - r would be 0, meaning N = r. But N has to be a three-digit number, and r is a single-digit remainder, so m cannot be 0. We're looking for m >= 1.

Now, let's rearrange this equation to directly express N:

N = 168m + r

This, my friends, is our general form! It’s the blueprint for every single number that satisfies the conditions of giving the same remainder r when divided by 8, 12, and 21. It beautifully encapsulates all the divisibility requirements and the common remainder into one concise expression. This form is incredibly useful because it allows us to systematically generate candidates for N by simply varying m (our positive integer multiplier for the LCM) and r (our valid non-zero remainder). Instead of randomly picking numbers and checking them, we now have a highly efficient formula. We also know the valid values for r from our previous section: r ∈ {1, 2, 3, 4, 5, 6, 7}. This combination of the general form and the restricted range for r is the backbone of our problem-solving strategy. It transforms a seemingly complex search into a structured exploration, ensuring we don't miss any potential three-digit numbers while efficiently ruling out those that don't fit our specific criteria. This is truly where the beauty of number theory shines, providing elegant tools to navigate the vast landscape of numbers!

The Grand Hunt: Pinpointing the Smallest Three-Digit Number

Alright, the moment of truth! We have our powerful equation, N = 168m + r, and we know that m is a positive integer (m >= 1) and r is a non-zero remainder (r ∈ {1, 2, 3, 4, 5, 6, 7}). Now, let's apply the crucial constraint that N must be a three-digit number. This means N must be greater than or equal to 100 and less than or equal to 999. Mathematically, 100 <= N <= 999. We need to use this range to figure out the possible values for m.

Let's start by finding the smallest three-digit number that fits our criteria. To make N as small as possible, we should choose the smallest possible value for m and the smallest possible value for r. The smallest possible value for m is 1. The smallest possible value for r is 1.

So, let's substitute these into our equation: N_min = 168 * 1 + 1 N_min = 168 + 1 N_min = 169

Is 169 a valid three-digit number? Yes, it's clearly between 100 and 999. Let's double-check if it satisfies all the remainder conditions:

  • 169 / 8: 169 = 8 * 21 + 1. The remainder is 1.
  • 169 / 12: 169 = 12 * 14 + 1. The remainder is 1.
  • 169 / 21: 169 = 21 * 8 + 1. The remainder is 1.

Bingo! All divisions yield the same non-zero remainder of 1. This confirms that 169 is indeed the smallest three-digit natural number that satisfies all the problem's conditions. This systematic approach, leveraging our derived general form and understanding the bounds for m and r, allows us to pinpoint the answer with certainty. We didn't just guess; we used solid mathematical logic to arrive at this specific number. This illustrates the elegance of applying number range constraints to an optimization problem, finding the minimum value efficiently. It's truly satisfying to see how these abstract mathematical concepts come together to solve a concrete problem!

The Final Frontier: Discovering the Largest Three-Digit Number

Alright, math adventurers, we've conquered the smallest number, and now our sights are set on the largest three-digit number that meets our challenging criteria. Just like before, we'll lean on our trusty general form: N = 168m + r. We know r can be any value from 1 to 7. To find the largest possible N, we need to maximize both m and r within the three-digit number range (100 <= N <= 999).

First, let's figure out the upper limit for m. Our number N must be N <= 999. So, 168m + r <= 999. To find the maximum possible m, let's consider the smallest possible r for a moment, which is 1. 168m + 1 <= 999 168m <= 998 m <= 998 / 168 m <= 5.94...

Since m must be an integer, the largest possible integer value for m is 5. If m were 6, then 168 * 6 = 1008, which is already a four-digit number, even before adding r. So, m definitely cannot exceed 5. This crucial step in number range optimization helps us define the upper boundary for m effectively.

Now that we know the maximum m is 5, to get the largest possible N, we should combine this with the largest possible remainder r. The largest possible r is 7.

Let's plug these maximum values into our equation: N_max = 168 * 5 + 7 N_max = 840 + 7 N_max = 847

Is 847 a valid three-digit number? Absolutely, it falls squarely between 100 and 999. Let's perform our final sanity check, ensuring 847 indeed yields a remainder of 7 when divided by 8, 12, and 21:

  • 847 / 8: 847 = 8 * 105 + 7. Remainder is 7.
  • 847 / 12: 847 = 12 * 70 + 7. Remainder is 7.
  • 847 / 21: 847 = 21 * 40 + 7. Remainder is 7.

Fantastic! All conditions are met. 847 is officially the largest three-digit natural number that satisfies all the problem’s requirements. This journey from deciphering the initial conditions to systematically applying LCM, understanding remainder properties, and then strategically navigating number ranges to maximize our output is a prime example of effective problem-solving skills in mathematics. It showcases how a solid understanding of fundamental number theory concepts empowers us to tackle seemingly complex challenges with confidence and precision.

Wrapping It Up: Our Mathematical Journey's End

What a ride, guys! We started with a seemingly complex mathematical puzzle about three-digit numbers and consistent remainders, and through a systematic, step-by-step approach, we've not only solved it but also gained a deeper appreciation for the beauty and logic of number theory. We identified the problem's core, understood the critical role of the Least Common Multiple (LCM), meticulously calculated it, and established the precise limits for our non-zero remainder. From there, we constructed a powerful general formula, N = 168m + r, which became our guiding star. By carefully applying the three-digit number range (100 to 999) and strategically selecting the smallest and largest valid values for m and r, we successfully uncovered both the smallest three-digit number, which is 169, and the largest three-digit number, which is 847.

This wasn't just about getting two answers; it was about the journey of problem-solving. It highlighted how breaking down a problem into smaller, manageable parts, understanding the underlying mathematical concepts, and applying them with precision can demystify even the trickiest questions. These kinds of challenges are incredibly valuable because they build not just your numerical skills but also your logical reasoning and analytical abilities. So, next time you encounter a problem involving divisibility, remainders, or common multiples, remember the steps we took today. You've now got the tools and the confidence to tackle similar number theory problems head-on! Keep exploring, keep questioning, and most importantly, keep enjoying the fascinating world of mathematics!