HMMT 2017 T7

Let $p$ be a prime. A complete residue class modulo $p$ is a set containing at least one element equivalent to $k(\bmod p)$ for all $k$. (a) (20) Show that there exists an $n$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$. (b) (25) Show that there exists an $n \leq p^{2}$ such that the $n$th row of Pascal's triangle forms a complete residue class modulo $p$.