For how many primes $p$ less than $15$, there exists integer triples $(m,n,k)$ such that
\[
\begin{array}{rcl}
m+n+k &\equiv& 0 \pmod p \\
mn+mk+nk &\equiv& 1 \pmod p \\
mnk &\equiv& 2 \pmod p.
\end{array}
\]

$ 
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 6
$