I am having trouble understanding why the "Hence $p$ divides . . . " part follows.
This is from the Wiki article on Lucas's Theorem. Help appreciated!
If $p$ is a prime and $n$ is an integer with $1≤n≤p-1$, then the numerator of the binomial coefficient
$\binom p n = \frac{p \cdot (p-1) \cdots (p-n+1)}{n \cdot (n-1) \cdots 1}$ is divisible by $p$ but the denominator is not. Hence $p$ divides $\tbinom{p}{n}$.
Write your fraction as $a/b$. Since $p$ divides $a$, you can write $a=kp$ for some integer $k$, so your fraction is $kp/b$. It's an integer, so $b$ divides $kp$.
Now $p$ does not divide $b$, and it is prime, therefore $p$ and $b$ are coprime; Gauss's theorem allows you to conclude that $b$ divides $k$, and so your fraction is actually $p\times k/b$, where $k/b$ is an integer, and so it is divisible by $p$.
