On the Wikipedia page about Binomial coefficients, it states that $p$ divides $\binom{p}{k}$ where $p$ is prime. It offers a proof that I don't understand from the very first line. It says that it is true since $\binom{p}{k}$ is a natural number and $p$ divides the numerator but not the denominator. I don't get how that shows us anything really.
Please could you explain why the above statement is true?
Many thanks.
We know that $\binom pk = n = p \frac ab$ where $a, b$ are integers and $p$ is not a factor of $b$. Therefore, $pa=bn$ so $p \mid bn$. We also know, because $p$ does not divide $b$, that $p$ and $b$ are relatively prime. Therefore, it must be the case that $p$ divides the other factor of the product, which is $n = \binom pk$.