Question 160
For each positive integer $n$, let
$$
C_n = \frac{1}{n+1} \binom{2n}{n},
$$
where, as usual,
$$
\binom{2n}{n} = \frac{(2n)!}{(n!)^2}.
$$
(i) Prove that $C_n$ is an integer for every positive integer $n$.
(ii) Prove that $C_n$ is not a prime number if $n \gt 3$.
$$
C_n = \frac{1}{n+1} \binom{2n}{n},
$$
where, as usual,
$$
\binom{2n}{n} = \frac{(2n)!}{(n!)^2}.
$$
(i) Prove that $C_n$ is an integer for every positive integer $n$.
(ii) Prove that $C_n$ is not a prime number if $n \gt 3$.