Deduce that, for all $n \in \mathbb { N } ^ { * } , k \in \mathbb { N }$ and $p$ prime number: if $p ^ { k }$ divides $\binom { 2 n } { n }$, then $p ^ { k } \leqslant 2 n$.
Deduce that, for all $n \in \mathbb { N } ^ { * } , k \in \mathbb { N }$ and $p$ prime number: if $p ^ { k }$ divides $\binom { 2 n } { n }$, then $p ^ { k } \leqslant 2 n$.