I got a question, for a number theory assignment. But I don't understand what it's saying at all; here it is:
Fix $n>1 $ and for a prime $p$ , define $r(p)$ as the unique integer s.t $2^{r(p)} \leq 2n <2^{r(p)+1}$. Show that the exponent of $p$ in $n!$ is $\sum_i^{r(p)}[\tfrac{n}{p_i}]$ where [] denotes the integer part.
I have some specific questions, but I understand so poorly what this question is saying, that I will not be asking enough. So, please add more detail to help me understand, more than, just what I'm asking. But, here's a few of my specific questions:
1) What is meant by "the exponent of $p$ in $n!$"?
2) Does answering the question have something to do with using the giveninequalities in order to make an inequality for $n!$?
3) There was explicit dependence stated for $p$ in $r(p)$. So, how do you extract any information about it?