To find the last 5 non-zero digits of \(288!\), we must first remove trailing zeros caused by factors of 10, which are produced by multiplying factors of 2 and 5. After that, we need to compute the factorial and keep only the significant non-zero digits.

Let’s break this down step by step:

### 1. **Trailing Zeros Removal**:
   The number of trailing zeros in a factorial is determined by the number of factors of 5 in it. In this case, we need to compute the number of factors of 5 in \(288!\).

\[
\left\lfloor \frac{288}{5} \right\rfloor + \left\lfloor \frac{288}{25} \right\rfloor + \left\lfloor \frac{288}{125} \right\rfloor = 57 + 11 + 2 = 70
\]

So, \(288!\) has 70 trailing zeros, meaning we can remove these zeros after computing the factorial.

### 2. **Modulo Approach**:
   Instead of calculating \(288!\) directly (which is computationally very large), we use a method that multiplies only significant digits modulo a large power of 10, such as \(10^5\), skipping over the factors of 10 (i.e., factors of 2 and 5 that produce zeros).

Let me compute this to get the result.It seems the computation took too long. Let me refine the process for efficiency and compute the last five non-zero digits of \(288!\) using a faster approach.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can still guide you through the steps to compute the last 5 non-zero digits of \(288!\) manually or explain the approach in more detail. Would you like to proceed with that?

Here are 5 related questions that could expand on this topic:
1. How are trailing zeros in factorials calculated?
2. What is the role of prime factorization in computing factorials?
3. How can the modular arithmetic method simplify large factorial computations?
4. What techniques exist for dealing with very large numbers in programming?
5. How would the method change for finding non-zero digits in powers or products?

**Tip:** When computing large factorials, breaking down the problem using modular arithmetic can make the calculations more manageable.

Learn how to calculate the last 5 non-zero digits of 288! by using prime factorization, trailing zero removal, and modulo arithmetic. This problem illustrates key concepts in advanced math, including factorials and handling large numbers efficiently. Suitable for high school students in Grades 10-12.

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