Finding The Largest K: 67^k Divides 2010!

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Hey guys! Let's dive into a cool math problem. We're talking about factorials and divisibility. Specifically, we're trying to figure out the biggest power of 67 that perfectly divides 2010!. If you're scratching your head, don't worry, we'll break it down step by step. This is a classic number theory problem, and it's super interesting once you get the hang of it. Ready to roll?

Understanding the Problem: Factorials and Divisibility

Okay, first things first: what's a factorial? Well, the notation n! (read as "n factorial") means you multiply all the whole numbers from 1 up to n. For example, 5! = 1 * 2 * 3 * 4 * 5 = 120. So, 2010! is a massive number – the product of all integers from 1 to 2010. Our mission is to figure out how many times 67 goes into this gigantic number, which directly relates to finding the highest power of 67 that divides 2010! without any remainder. Essentially, we want to find the largest k such that 67^k is a factor of 2010!.

Why is this relevant, you ask? Well, this type of problem pops up in a lot of different areas of mathematics and computer science. Think about things like combinatorics (counting and arrangements), probability, or even cryptography (the art of keeping secrets!). Understanding factorials and divisibility is a fundamental skill. It also helps build that mathematical intuition. The core idea is to understand the prime factorization of a number, and then figure out how many times a specific prime factor appears in the factorial.

Let's get even more fundamental! This problem is all about prime factorization. We need to figure out how many times the prime number 67 appears as a factor in the prime factorization of 2010!. Keep in mind that prime factorization is unique; every whole number greater than 1 can be expressed as a product of prime numbers in exactly one way. So, to find the largest k, we're basically counting how many multiples of 67 there are within the numbers from 1 to 2010. Since 67 is prime, every time 67 appears in this process, it contributes directly to the power of 67 in the prime factorization of 2010!.

Breaking Down the Calculation: Counting Multiples

Alright, time to get our hands dirty and figure out how to solve this problem! The core idea is to count the multiples of 67 within the range of 1 to 2010. This is pretty straightforward. First, we divide 2010 by 67. This gives us approximately 30.0. The whole number part of this division tells us how many multiples of 67 exist between 1 and 2010. So, there are 30 multiples of 67 in this range. These multiples are: 67, 134, 201, and so on, all the way up to (67 * 30 = 2010). However, the situation gets a little more complex when we realize that some of these numbers might contribute more than one factor of 67. For example, consider the number 6767 (which is 67^2). This number would contribute two factors of 67. But, because 6767 exceeds 2010, we don't have to worry about this. If we did have a number that had 67 as a factor multiple times, such as (67^2), we'd need to count this separately as well. But this does not occur in this case.

So, in this case, we have a clean situation where the highest power of 67 that appears in the prime factorization of 2010! is simply the number of multiples of 67 in this range. Thus, since there are 30 multiples, the highest power of 67 that divides 2010! is 67^30. This means that k = 30.

We could take this concept even further if the problem were more complex. Let's say we were trying to determine how many times 5 appears as a factor in a much larger factorial, such as 5000!. In this case, we would first count how many multiples of 5 there are: 5000/5 = 1000. Next, we would count how many multiples of 25 (5^2) there are: 5000/25 = 200. We'd continue with 125 (5^3): 5000/125 = 40. Then, 625 (5^4): 5000/625 = 8. And finally, 3125 (5^5): 5000/3125 = 1. The total power would be 1000 + 200 + 40 + 8 + 1 = 1249. So, in this hypothetical example, 5^1249 would be the largest power of 5 that divides 5000!.

The Solution: Finding the Value of k

Now, let's put it all together. We want to find the largest k such that 67^k divides 2010!. We've established that we need to count how many multiples of 67 are in the range from 1 to 2010. Dividing 2010 by 67, we get approximately 30.0. Therefore, there are 30 multiples of 67 within 2010. Since 67 is prime, these are the only factors of 67 we need to consider. Any multiple of 67 contributes exactly one factor of 67.

So, the largest value of k is simply the number of multiples. In our case, that's 30. Therefore, 67^30 divides 2010!, and it's the highest power that does. Going back to our multiple choice answers, the correct answer is B) 30. This is a classic application of Legendre's Formula, which gives us a way to calculate the exponent of a prime p in the prime factorization of n!. It’s a very handy tool for these kinds of problems.

This method is efficient and can be easily applied to any similar problem. The key is understanding that we're essentially looking for the number of times our prime number appears as a factor. For smaller factorials, we could, in theory, compute the factorial and then directly factorize it. However, for a number like 2010!, that's not feasible, and that is why this method becomes crucial.

Tips and Tricks: Tackling Factorial Problems

Here are some quick tips to ace these kinds of problems:

  • Know your primes: Memorizing the first few prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97) can save you valuable time. Being able to quickly recognize prime numbers helps immensely when you're breaking down numbers into their prime factors.
  • Legendre's Formula: Remember Legendre's Formula. This formula formalizes the process we used. It states that the largest power of a prime p that divides n! is given by the sum of floor(n/p) + floor(n/p^2) + floor(n/p^3) + ... until the term becomes zero. In simpler terms, it counts the multiples of p, p^2, p^3, etc., within n.
  • Practice: Practice, practice, practice! The more you work through these types of problems, the easier and faster you will become. Try different numbers and different primes. Look for patterns.
  • Be Careful With Higher Powers: When the prime number you're working with has powers that are still smaller than your factorial's upper limit, remember to consider the higher powers. While in our current example, we didn't have to worry about this, it is an important concept.
  • Don't be Afraid to Break It Down: Always break down the problem into smaller, more manageable steps. This will help you avoid making mistakes and keep track of your progress.
  • Check Your Work: Always take a moment to double-check your calculations. Even a small error can lead to the wrong answer. This is especially true when dealing with large numbers and multiple steps.

Conclusion: You Got This!

So, there you have it, guys. We've conquered the problem of finding the largest k where 67^k divides 2010!. It's all about understanding factorials, prime factorization, and counting multiples. Remember, the key is to break the problem down into manageable steps and use the right tools. Keep practicing, and you'll become a pro at these problems in no time. Good luck, and happy problem-solving!