In my text book there's an example of a proof on why $\binom{p}{j}$ is divisible by $p$, with $p$ prime, for $0<j<p$. Firstly, it shows that
$$\binom{p}{j}=p\frac{(p-1)!}{j!(p-j)!}$$
From this it also points out that we know $p-1, j, p-j$ are all less than $p$, and therefore $p$ is not in the prime factorization of neither of them. Then it says (I quote): "Now, from the identity of equation 26..." (the one written above) "$p$ must be a factor of $\binom{p}{j}$".
Two correlated doubts: why do we need to note that $p$ does not appear in the prime factorization of $j$, $p-1$ and $p-$j? And why is it not enough to show that $\binom{p}{j}=pk$, as it is shown on the equation, to demonstrate that $\binom{p}{j}$ is divisible by $p$?