Finding 'n': Divisibility Rule Of 21 Explained

Hey guys! Let's dive into a cool math problem. We're gonna figure out the value of 'n' when we know that 4n is perfectly divisible by 21. Sounds fun, right? This isn't just about crunching numbers; it's about understanding how divisibility works and applying that knowledge to solve a puzzle. So, grab your pencils, and let's get started. We'll break down the concepts, step by step, making sure everyone understands the process. This is all about the joy of problem-solving. It's about taking a seemingly complex problem and breaking it down into manageable chunks. By the end of this, you'll not only have the answer but also a better grasp of number theory.

First, let's talk about divisibility. What does it actually mean for a number to be 'divisible'? Basically, it means that when you divide one number by another, you get a whole number, with no remainders. For example, 10 is divisible by 2 because 10 divided by 2 equals 5, a whole number. However, 11 is not divisible by 2, because 11 divided by 2 equals 5.5, which isn't a whole number. When we say that 4n is divisible by 21, it means that 4n can be divided by 21 without leaving any leftovers. This is the cornerstone of our problem. Understanding this concept is absolutely key to unlocking the solution. So, keep this definition in your mind as we move forward! It’s all about finding that whole number relationship.

Decoding the Divisibility of 21

Alright, let's zoom in on the number 21. The prime factorization of 21 is 3 x 7. This is super important because it tells us that if a number is divisible by 21, it must also be divisible by both 3 and 7. Think of it like a secret code: to unlock the 21 door, you need the keys for both 3 and 7. This is a fundamental principle in number theory, and it's super handy when dealing with divisibility problems. So, if 4n is divisible by 21, it means 4n must be divisible by both 3 and 7. Therefore, to solve this problem effectively, we must first ensure 4n is divisible by both 3 and 7. The strategy involves breaking down the divisibility into simpler checks using the prime factors.

Now, let's consider divisibility by 3. A number is divisible by 3 if the sum of its digits is divisible by 3. This is a neat trick, isn’t it? For example, 12 is divisible by 3 because 1 + 2 = 3, and 3 is divisible by 3. Similarly, 123 is divisible by 3 because 1 + 2 + 3 = 6, and 6 is divisible by 3. Remember, we are examining 4n, not just n. The number 4 will not affect the divisibility by 3 unless 'n' makes the total sum of digits of 4n a multiple of 3. Next, let’s look at the divisibility of 7. There's a rule for divisibility by 7 too, although it is a bit more involved. Double the last digit of the number and subtract it from the remaining truncated number. If the result is divisible by 7, then the original number is also divisible by 7. Let's see how this works! For example, take 49. Double the last digit (9 x 2 = 18). Subtract 18 from 4 (the remaining digits) , we get 4 - 18 = -14. Since -14 is divisible by 7, 49 is divisible by 7. Understanding and applying these rules helps us find the value of n.

Solving for 'n': The Step-by-Step Approach

Okay, now let's apply this knowledge to find the value of 'n'. We know that 4n must be divisible by both 3 and 7. We can consider different values of n and check whether the result, 4n, is divisible by 21. Let's start trying some small numbers. If n = 1, then 4n = 4. Is 4 divisible by 21? Nope. If n = 2, then 4n = 8. Still a no-go. Let’s keep going. If n = 3, then 4n = 12. No again. Keep in mind that we're looking for a number that, when multiplied by 4, gives us a result divisible by both 3 and 7. This process involves testing various potential values for 'n' to pinpoint the one that satisfies the condition of 4n being divisible by 21. The most important thing here is to be patient and systematic. Do not give up if your first few tries don’t work out. Remember to ensure that 4n satisfies the criteria for both 3 and 7, which we discussed earlier, through our prime factorization of 21.

Let’s try a few more. If n = 4, then 4n = 16, which is still not divisible by 21. And if n = 5, then 4n = 20, also not divisible by 21. What happens when we try n = 6? Then 4n = 24. No dice. But wait, if n = 7, then 4n = 28. If n = 8, then 4n = 32. If n = 9, then 4n = 36. Getting closer... Finally, if n = 10, then 4n = 40. Now, let’s think about this a bit more strategically. Since 4n must be divisible by 21, it must also be a multiple of 21. So, let’s think about the multiples of 21: 21, 42, 63, 84, and so on. We are looking for a multiple of 21 that can be expressed as 4n. Therefore, we should check which of these numbers can be divided by 4 without any remainders. The easiest way to find a suitable value for 'n' is to identify a multiple of 21 that is also divisible by 4. So, let’s go through those multiples we just discussed. First, 21 is not divisible by 4. Then, 42 is not divisible by 4. However, 63 is not divisible by 4 either, but 84 is. We can divide 84 by 4, and the answer is 21! That means 4n = 84. Therefore, 'n' must be 21.

Final Answer and Conclusion

So, the answer is n = 21. If you put n = 21 into our original equation, 4n becomes 4 * 21 = 84. And guess what? 84 is indeed divisible by 21. We have successfully found the value of 'n' that satisfies the conditions of the problem! Isn’t that amazing? We broke down a seemingly complex problem into simple steps, applied divisibility rules, and systematically solved it. This isn't just about getting the right answer; it's about the journey of understanding and learning. Remember, the core of this problem lies in understanding divisibility rules and prime factorization. So, the next time you come across a divisibility question, you'll be well-prepared to tackle it. This process gives you the confidence to approach similar problems with a systematic and logical approach. We used our knowledge of prime factors to ensure divisibility by 21 and the step-by-step method helped us find a logical answer. Great job, everyone! Keep practicing, and you’ll become a math whiz in no time. Always remember that math is like a puzzle: the more you play with it, the better you get!