Number Theory


Factorials: Level 3 Challenges


How many of the following 99 integers are primes?

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

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

Find the remainder when 70!70! is divided by 51835183.

Note: Don't use a computational device!

limn(2nn)1n= ?\large \lim_{n \rightarrow \infty} { 2n \choose n } ^ { \frac{1}{n} } = \ ?

How many trailing zeroes are in the decimal representation of n=1+k=12013k!(k3+2k2+3k+1)?n=1+\displaystyle{\sum_{k=1}^{2013}k!(k^3+2k^2+3k+1)}?


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