Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$
$\textbf{(A) } 4444 \qquad\textbf{(B) } 4494 \qquad\textbf{(C) } 4944 \qquad\textbf{(D) } 9444 \qquad\textbf{(E) } 9944$
For a number to be divisible by $4,$ the last two digits have to be divisible by $4.$ That means the last two digits of this integer must be $4.$
For a number to be divisible by $9,$ the sum of all the digits must be divisible by $9.$ The only way to make this happen is with nine $4$'s. However, we also need one $9.$
The smallest integer that meets all these conditions is $4444444944$. The last four digits are $\boxed{\mathrm{(C) \ } 4944}$