The homework problem asks for "the maximum power of a number $n$ such that $n$ divides 500! (500 factorial)." To clarify, you're tasked with finding the maximum power of some integer $n$ that divides $500!$, meaning you need to determine how many times $n$ divides evenly into $500!$. This type of problem is typically solved using prime factorization.

If the problem is asking for the maximum power of a specific prime number $p$ (or another integer), you'll need to find how many times $p$ can divide into $500!$. This is calculated by:

$\text{Maximum Power of } p = \sum_{k=1}^{\infty} \left\lfloor \frac{500}{p^k} \right\rfloor$ where $\left\lfloor x \right\rfloor$ denotes the floor function, which rounds down to the nearest integer.

For example, if the number $n$ is a prime like 2, 3, 5, etc., you'd calculate this sum for the chosen $p$.

Let me know:

• Is there a specific number $n$ for which you need to find the maximum power in $500!$?

Or I can guide you through a general example!

Would you like further explanation, or should I solve an example for a specific prime like 2, 3, or 5?

Related questions to expand on the topic:

1. How do you calculate the maximum power of a prime factor in any factorial?
2. What is the purpose of the floor function in this type of problem?
3. Can this method be extended to composite numbers like 12 or 20 instead of primes?
4. What other real-life applications does prime factorization have?
5. How does the rate of decrease in the quotient $\frac{500}{p^k}$ influence the calculation?

Tip:

In such problems, prime factorization is key. Always break the problem into smaller parts to handle large numbers systematically.