How many zeros are there in the end of the numbers 200! ?

To determine how many zeros are at the end of \(200!\), we need to count the number of factors of 10 in \(200!\). Since each factor of 10 is the product of a factor of 2 and a factor of 5, the number of trailing zeros is determined by the number of factors of 5 (because there are always more factors of 2 than factors of 5 in factorials).

The number of factors of 5 in \(200!\) can be found by summing:

\[
\left\lfloor \frac{200}{5} \right\rfloor + \left\lfloor \frac{200}{25} \right\rfloor + \left\lfloor \frac{200}{125} \right\rfloor
\]

This counts how many multiples of 5, 25, and 125 there are in \(200!\).

Now let's calculate each term:

\[
\left\lfloor \frac{200}{5} \right\rfloor = 40, \quad \left\lfloor \frac{200}{25} \right\rfloor = 8, \quad \left\lfloor \frac{200}{125} \right\rfloor = 1
\]

Adding these together:

\[
40 + 8 + 1 = 49
\]

Therefore, the number of trailing zeros in \(200!\) is **49**.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How many zeros are at the end of \(100!\)?
2. What is the prime factorization of \(200!\)?
3. How does the number of trailing zeros change for factorials of powers of 10 (e.g., \(1000!\))?
4. How would the number of trailing zeros change if we considered base 8 instead of base 10?
5. How many digits are there in \(200!\)?

**Tip:** The formula used to count trailing zeros can be generalized to other bases by considering the prime factors of the base.

How many zeros are there in the end of the numbers 200!?

Learn how to calculate the number of trailing zeros in 200!. Using number theory, this problem explores how to find trailing zeros in factorials by counting factors of 5. Find out why 200! has 49 trailing zeros. Suitable for high school students in Grades 9-12.

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