Find the number of zeros at the end of 10! when written in base 3.

To find the number of zeros at the end of \(10!\) (10 factorial) when it is written in base 3, we need to consider how many times 3 is a factor in \(10!\).

### Steps to Solve:

1. **Factorize 10!:**
   \[
   10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
   \]
   
2. **Count the number of factors of 3 in \(10!\):**
   We need to determine how many times 3 is a factor in each of the numbers from 1 to 10.

   - \(3\) is a factor in: \(3\) (once), \(6\) (once), and \(9\) (twice, since \(9 = 3^2\)).
   
   - So, the total number of times 3 appears as a factor in \(10!\) is:
   \[
   1 + 1 + 2 = 4
   \]
   
3. **Number of zeros at the end of \(10!\) in base 3:**
   The number of zeros at the end of a number in base 3 is determined by the highest power of 3 that divides \(10!\). Since we have four factors of 3 in \(10!\), there are **4 zeros** at the end of \(10!\) when written in base 3.

So, the number of zeros at the end of \(10!\) in base 3 is **4**.

Would you like more details on this process or have any questions?

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Here are 8 related questions:
1. How do you find the number of trailing zeros in \(n!\) for different bases?
2. What is the significance of trailing zeros in different bases?
3. How do you factorize a factorial number to find specific prime factors?
4. How does changing the base affect the number of trailing zeros in \(n!\)?
5. Can you find the number of trailing zeros in \(15!\) when written in base 5?
6. What is the highest power of 2 that divides \(20!\)?
7. How do trailing zeros relate to the factorization of numbers?
8. What are the general steps for converting a factorial to any base?

**Tip:** To find the number of trailing zeros in a number, factorize the number and focus on the factors that contribute to the base you're considering.

Learn how to find the number of zeros at the end of 10 factorial (10!) when it is written in base 3. Understand the process of factorizing factorial numbers, counting specific prime factors like 3, and determining the number of trailing zeros in different bases. Suitable for advanced high school students studying number theory and factorials.

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