<p>Let <img alt="$N$" height="12" src="https://latex.artofproblemsolving.com/f/c/9/fc97ef67268cd4e91bacdf12b8901d7036c9a056.png" width="16" /> denote the number of ordered triples of positive integers <img alt="$(a, b, c)$" height="18" src="https://latex.artofproblemsolving.com/d/8/0/d80414f2f94f70731e91b6e12c4e6934c8fa65de.png" width="54" /> such that <img alt="$a, b, c \leq 3^6$" height="20" src="https://latex.artofproblemsolving.com/f/4/6/f46754c99fa52b98837f384d3d62a93f82e48ae6.png" width="84" /> and <img alt="$a^3 + b^3 + c^3$" height="16" src="https://latex.artofproblemsolving.com/9/a/7/9a70858379029b66b2607e3f0fccbd8c76f1d648.png" width="91" /> is a multiple of <img alt="$3^7$" height="15" src="https://latex.artofproblemsolving.com/b/0/7/b07648d6e77e40902829c261a2e6d8e1a897e4bb.png" width="15" />. 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: 331

Let $N$ denote the number of ordered triples of positive integers $(a, b, c)$ such that $a, b, c \leq 3^6$ and $a^3 + b^3 + c^3$ is a multiple of $3^7$ . Find the remainder when $N$ is divided by $1000$ .

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Tags: Algebra Combinatorics

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