VN-TST 2017 Problem 2

For each positive integer $n$ , set $x_n=\binom{2n}{n}$ . a. Prove that if $\frac{2017^k}{2}<n<2017^k$ for some positive integer $k$ then $2017$ divides $x_n$ . b. Find all positive integer $h>1$ such that there exists positive integers $N,T$ such that $(x_n)_{n>N}$ is periodic mod $h$ with period $T$ .