MathNet 2025-part-02 — 186 problems (EN–VI) Question 138

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