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The question in the image asks:

**"What is the maximum power of n such that n divides 500! (500 factorial)?"**

This is a typical problem of finding the largest exponent \( k \) such that a number \( n^k \) divides \( 500! \).

### Step-by-Step Solution:

Let’s assume \( n \) is a prime number. To find the highest power \( k \) of a prime \( n \) that divides \( 500! \), we use **Legendre’s formula** (also called de Polignac's formula):

\[
k = \sum_{i=1}^{\infty} \left\lfloor \frac{500}{n^i} \right\rfloor
\]

Where \( \left\lfloor x \right\rfloor \) denotes the greatest integer less than or equal to \( x \), and \( n^i \) is the successive powers of the prime \( n \).

The steps are:
1. Divide 500 by \( n \) (the prime number).
2. Divide the result by \( n \) again (repeating until the division is zero).
3. Sum all the quotients (rounded down).

If \( n \) is a composite number, first factorize it into primes and apply the formula to each prime factor separately.

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If you provide a specific value for \( n \), I can compute the answer for that prime or composite number.

Would you like more details or help with a specific value of \( n \)?

### Follow-Up Questions:
1. How is Legendre’s formula used in prime factorization of factorials?
2. What is the highest power of 2 that divides 500!?
3. Can we apply the same method to non-prime values of \( n \)?
4. What role do prime numbers play in this type of divisibility problem?
5. How would the solution change if the factorial were a different number, like 1000!?

**Tip:** For composite numbers, always break them down into their prime factors before applying Legendre’s formula!

What is the maximum power of n such that n divides 500! (500 factorial)?

Solve the problem of finding the maximum power of n such that it divides 500 factorial (500!). Using Legendre's formula, this step-by-step solution guides you through prime factorization and calculating the largest exponent k for divisibility. Suitable for high school math students.

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