Let's solve these two problems one by one:

1. Prime Factorization of $10!$

To find the prime factorization of $10!$, we first compute $10!$.

$10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

Now, breaking each number into its prime factors:

• $10 = 2 \times 5$
• $9 = 3 \times 3$
• $8 = 2 \times 2 \times 2$
• $7 = 7$
• $6 = 2 \times 3$
• $5 = 5$
• $4 = 2 \times 2$
• $3 = 3$
• $2 = 2$
• $1 = 1$ (which has no prime factors)

Now, combining all these prime factors together:

$10! = 2^8 \times 3^4 \times 5^2 \times 7$

So, the prime factorization of $10!$ is:

$\boxed{2^8 \times 3^4 \times 5^2 \times 7}$

2. Number of Zeros at the End of $100!$

To find the number of zeros at the end of $100!$, we need to count the number of factors of 10 in $100!$. Each factor of 10 comes from a pair of 2 and 5. Since there are always more factors of 2 than 5 in a factorial, we only need to count the number of factors of 5.

The number of factors of 5 in $100!$ is calculated using the formula:

$\left\lfloor \frac{100}{5} \right\rfloor + \left\lfloor \frac{100}{25} \right\rfloor$

First, calculate:

$\left\lfloor \frac{100}{5} \right\rfloor = 20$

$\left\lfloor \frac{100}{25} \right\rfloor = 4$

Now, adding them together:

$20 + 4 = 24$

Thus, the number of zeros at the end of $100!$ is $\boxed{24}$.

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Would you like further details or have any questions about these solutions? Here are 5 related questions for your interest:

1. How do you calculate the prime factorization of a larger factorial like $20!$?
2. Why do we only count the factors of 5 when determining the number of trailing zeros?
3. How would you calculate the number of zeros at the end of $50!$?
4. Can the number of trailing zeros be zero for smaller factorials, and if so, why?
5. What is the prime factorization of $15!$?

Tip: When calculating trailing zeros, always remember that a zero at the end is created by a factor of 10, which is the product of 2 and 5. Hence, the number of trailing zeros is determined by the limiting factor—usually the number of 5s in the factorial's prime factorization.