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Factorials

2! = 2, 3! = 3*2, 4! = 4*3*2… and 100! is a lot better than writing out 158 digits. 89! is the largest factorial that can fit in a tweet.

Level 3

         

How many of the following 99 integers are primes?

\[\begin{array} &100!+2, &100!+3, &100!+4, &\ldots, &100!+100\end{array}\]

How many positive integers are divisors of \(21!\) but are not divisors of \(20!\)?

Find the remainder when \(70!\) is divided by \(5183\).

Note: Don't use a computational device!

\[\large \lim_{n \rightarrow \infty} { 2n \choose n } ^ { \frac{1}{n} } = \ ?\]

How many trailing zeroes are in the decimal representation of \[n=1+\displaystyle{\sum_{k=1}^{2013}k!(k^3+2k^2+3k+1)}?\]

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