The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is $A?$
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 9$
First, modulo $2$ or $5$, $\underline{20210A} \equiv A$.
Hence, $A \neq 0, 2, 4, 5, 6, 8$.
Second modulo $3$, $\underline{20210A} \equiv 2 + 0 + 2 + 1 + 0 + A \equiv 5 + A$.
Hence, $A \neq 1, 4, 7$.
Third, modulo $11$, $\underline{20210A} \equiv A + 1 + 0 - 0 - 2 - 2 \equiv A - 3$.
Hence, $A \neq 3$.
Therefore, the answer is $\boxed{\textbf{(E)}\ 9}$.
~NH14 ~Steven Chen (www.professorchenedu.com)