<p>For each positive integer <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" />, let <img alt="$f(n)$" height="18" src="https://latex.artofproblemsolving.com/e/e/f/eef0203e65664bd00b5f0325c5c5fcd81bcead8d.png" width="34" /> be the sum of the digits in the base-four representation of <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> and let <img alt="$g(n)$" height="18" src="https://latex.artofproblemsolving.com/3/f/d/3fda49917e5cf30f3fdcf2fbf4ed52d62ab10fbf.png" width="33" /> be the sum of the digits in the base-eight representation of <img alt="$f(n)$" height="18" src="https://latex.artofproblemsolving.com/e/e/f/eef0203e65664bd00b5f0325c5c5fcd81bcead8d.png" width="34" />. For example, <img alt="$f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$" height="18" src="https://latex.artofproblemsolving.com/7/2/e/72e74614420fa6e6adf0c1d11e4c702c4af50358.png" width="263" />, and <img alt="$g(2020) = \text{the digit sum of }12_{\text{8}} = 3$" height="18" src="https://latex.artofproblemsolving.com/f/c/8/fc854828c27b0b0f5a058ded8f2584f7aadb60ce.png" width="264" />. Let <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> be the least value of <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> such that the base-sixteen representation of <img alt="$g(n)$" height="18" src="https://latex.artofproblemsolving.com/3/f/d/3fda49917e5cf30f3fdcf2fbf4ed52d62ab10fbf.png" width="33" /> cannot be expressed using only the digits <img alt="$0$" height="12" src="https://latex.artofproblemsolving.com/b/c/1/bc1f9d9bf8a1b606a4188b5ce9a2af1809e27a89.png" width="8" /> through <img alt="$9$" height="12" src="https://latex.artofproblemsolving.com/b/f/2/bf2c9074b396e3af0dea52d792660eea1c77f10f.png" width="8" />. Find the remainder when <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> is divided by <img alt="$1000$" height="13" src="https://latex.artofproblemsolving.com/6/e/e/6ee927e1332358c96c62c277441c907c4f51057f.png" width="35" />.</p>

Question No: 373

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, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$ , and $g(2020) = \text{the digit sum of }12_{\text{8}} = 3$ . 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 $1000$ .

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Tags: Algebra

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