KöMaL 2021 A813

Let $p$ be a prime number and $k$ a positive integer. Let $t=\sum_{j=0}^{\infty} \left\lfloor\frac{k}{p^j}\right\rfloor.$ $a)$ Let $f(x)$ be a polynomial of degree $k$ with integer coefficients such that its leading coefficient is 1 and its constant is divisible by $p$. Prove that there exists $n\in \mathbb{N}$ for which $p\mid f(n)$, but $p^{t+1}\nmid f(n)$. $b)$ Prove that the statement above is sharp, i.e. there exists polynomial $g(x)$ of degree $k$, integer coefficients, leading coefficient 1 and constant divisible by $p$ such that if $p\mid g(n)$ is true for a certain $n\in \mathbb{N}$, then $p^t\mid g(n)$ also holds.