Question 5
For each positive integer $n,$ let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n).$ For example, $f(2020) = f(133210_4) = 10$ $= 12_8,$ and the digit sum of $12_8$ is $3,$ so $g(2020) = 3.$ Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9.$ Find the remainder when $N$ is divided by $1000.$