Find Numbers Divisible By 3 & 4: 91-164 Range

Hey everyone, ever wondered how to effortlessly figure out how many numbers within a specific range are divisible by not just one, but two different numbers? Well, guys, you're in for a treat because today we're diving deep into a super practical math problem that pops up more often than you'd think! We're going to tackle the question: How many numbers between 91 and 164 are perfectly divisible by both 3 and 4? This isn't just about finding an answer; it's about understanding the core concepts that empower you to solve similar problems with confidence and ease. Think of it as gaining a new superpower for your numerical toolkit! We'll break down the seemingly complex into easily digestible, friendly chunks, ensuring that by the end of this, you'll be a total pro at these types of calculations. Many people stumble upon divisibility questions and feel overwhelmed, but I promise you, with the right approach and a clear understanding of the rules, you'll find them not only manageable but actually enjoyable. We'll cover everything from the basic rules of divisibility by individual numbers to the magic of the Least Common Multiple (LCM), and then apply all that knowledge directly to our problem. So, grab a cup of coffee, get comfy, and let's unlock these mathematical secrets together, making what seems like a daunting task feel like a fun puzzle!

Decoding Divisibility Rules: A Deep Dive

Before we can even begin to think about numbers divisible by both 3 and 4, we first need to get a rock-solid grasp on what it means for a number to be divisible by each of these individually. Trust me, guys, knowing these fundamental rules isn't just about answering a math problem; it's about building a strong foundation for all sorts of numerical thinking. These aren't just obscure tricks; they are efficient mental shortcuts that can save you a ton of time and effort, whether you're balancing your checkbook, planning a party, or even coding. Let's peel back the layers and truly understand the why behind these handy rules.

Mastering Divisibility by 3

Alright, let's kick things off with one of the most elegant divisibility rules out there: the rule for 3. The rule states that a number is divisible by 3 if the sum of its digits is divisible by 3. Isn't that neat? You don't actually have to perform the long division to check; just add up the digits! For instance, take the number 123. The sum of its digits is 1 + 2 + 3 = 6. Since 6 is divisible by 3 (6 ÷ 3 = 2), then 123 must also be divisible by 3 (123 ÷ 3 = 41). See? Simple, effective, and quite powerful! Let's try another one, say 587. The sum of its digits is 5 + 8 + 7 = 20. Is 20 divisible by 3? Nope! (20 ÷ 3 leaves a remainder of 2). So, 587 is not divisible by 3. You can even apply this recursively for larger sums. If you get a large sum, just sum its digits again! For example, for 7893, the sum is 7+8+9+3 = 27. Since 27 is divisible by 3 (27 ÷ 3 = 9), then 7893 is divisible by 3. This rule works because of modular arithmetic, specifically how numbers behave in base 10. Any number can be written as a sum of its digits multiplied by powers of 10. Since 10 leaves a remainder of 1 when divided by 3 (10 = 3*3 + 1), any power of 10 also leaves a remainder of 1 when divided by 3. This mathematical property means that when you consider a number modulo 3, it's equivalent to considering the sum of its digits modulo 3. It’s like magic, but it’s just solid mathematics! Understanding this rule truly empowers you to quickly assess numbers without needing a calculator or tedious manual division. It's a foundational skill for anyone looking to sharpen their number sense and streamline their arithmetic processes. Keep practicing this one, guys; it's a real game-changer.

Unlocking Divisibility by 4

Next up, we have the rule for divisibility by 4, which is equally clever and super handy. The rule for 4 is that a number is divisible by 4 if the number formed by its last two digits is divisible by 4. That's right, you only need to look at the tail end of the number! This is fantastic because it means you don't care how big or small the number is; only the last two digits determine its divisibility by 4. Let's take 1,236 as an example. The number formed by its last two digits is 36. Is 36 divisible by 4? Yes, 36 ÷ 4 = 9. Therefore, 1,236 is divisible by 4 (1236 ÷ 4 = 309). How cool is that? Now, consider 5,178. The last two digits form the number 78. Is 78 divisible by 4? Well, 78 ÷ 4 is 19 with a remainder of 2. So, 5,178 is not divisible by 4. What about a number like 900? The last two digits form 00. Is 00 divisible by 4? Absolutely! 0 divided by any non-zero number is 0. So, 900 is divisible by 4. This rule works because 100 is divisible by 4 (100 ÷ 4 = 25). Any number can be expressed as a multiple of 100 plus its last two digits (e.g., 1236 = 12 * 100 + 36). Since the '100s part' is always divisible by 4, the divisibility of the entire number by 4 hinges solely on whether the remaining 'last two digits' part is also divisible by 4. This rule is particularly useful when dealing with larger numbers, where attempting full division would be time-consuming. It’s an essential shortcut for quick mental math and a key skill in problem-solving scenarios. Understanding this helps you see patterns in numbers and makes you much faster at calculations. So, next time you see a long number, just zero in on those final two digits to check for divisibility by 4!

The Power of LCM: When Multiple Rules Collide

Okay, guys, we've mastered divisibility by 3 and 4 individually. But our original question asks for numbers divisible by both 3 and 4. This is where a very special concept, the Least Common Multiple (LCM), comes into play. The LCM is your absolute best friend when you're looking for numbers that satisfy multiple divisibility conditions simultaneously. It helps us consolidate multiple rules into one streamlined check. Without understanding LCM, you'd be trying to apply two separate rules, which isn't wrong, but it's certainly not the most efficient way to solve these kinds of problems, especially when the numbers get larger or the conditions become more complex. The beauty of the LCM is that it gives us a single, combined divisibility rule that encompasses all the individual conditions. So, let's dive into why LCM is so crucial and how we calculate it.

Why LCM is Your Best Friend for "Both/And" Questions

When a number needs to be divisible by two or more different numbers, say 'a' and 'b', it essentially means that this number must be a multiple of 'a' and a multiple of 'b'. The smallest positive number that is a multiple of both 'a' and 'b' is their Least Common Multiple (LCM). If a number is a multiple of the LCM of 'a' and 'b', then it guarantees that the number is divisible by both 'a' and 'b' individually. Think about it: if a number is a multiple of LCM(a, b), it inherently contains all the prime factors of 'a' and all the prime factors of 'b', with their highest powers. This makes it divisible by both. For our specific problem, we need numbers divisible by both 3 and 4. So, we need to find LCM(3, 4). How do we do that? Since 3 and 4 are coprime (meaning they share no common prime factors other than 1), their LCM is simply their product. LCM(3, 4) = 3 × 4 = 12. This means any number that is divisible by both 3 and 4 must also be divisible by 12. And conversely, any number divisible by 12 will automatically be divisible by both 3 and 4. This simplification is incredibly powerful! Instead of checking two separate rules, we now only need to check one: is the number divisible by 12? This transformation is the core strategy for efficiently solving such problems. Imagine if we were checking for divisibility by 6 and 9. Their LCM is 18, so we'd only need to check for divisibility by 18. This principle extends to any set of numbers, making LCM an indispensable tool in number theory and practical arithmetic. It consolidates multiple conditions into a single, elegant rule, simplifying complex problems immensely. Mastering the LCM concept is truly a step up in your mathematical prowess, allowing you to tackle seemingly intricate problems with remarkable ease and speed.

Solving the Mystery: Numbers Between 91 and 164

Now that we know we're looking for numbers divisible by 12 (because LCM(3, 4) = 12), and our range is between 91 and 164, we can finally get down to brass tacks and solve the actual problem. This is where all our foundational knowledge comes together, guys. We need to find the first multiple of 12 that's greater than 91 and the last multiple of 12 that's less than 164. Once we have these two boundary numbers, counting the multiples in between becomes a straightforward task. It's like finding the first and last train stops within a specific segment of the track to count how many stations are there. This systematic approach ensures accuracy and clarity, preventing any missed numbers or accidental overcounting. Let’s break it down into manageable steps.

Pinpointing the First and Last Multiples

First, let's find the first multiple of 12 greater than 91. We can do this by dividing 91 by 12: 91 ÷ 12 = 7 with a remainder of 7 (since 12 × 7 = 84). This tells us that 84 is a multiple of 12, but it's less than 91. To get the next multiple, which will be the first one greater than 91, we simply add 12 to 84. So, 84 + 12 = 96. Thus, 96 is our first number that is divisible by both 3 and 4 within our desired range (it's between 91 and 164). This step is crucial because it sets our lower bound precisely. If we had chosen 84, it would be outside the