Cubes Summing to Multiples of 9

★ 7/10combinatorics number theory Counting Modular Arithmetic Casework

Problem

Let $M = \{1, 2, \dots, 10\}$ and

A = \left\{(x, y, z) : x, y, z \in M,\ 9 \mid x^3 + y^3 + z^3\right\}.

How many elements does $A$ have?

Answer

Solution

Difficulty	7/10
Topics	combinatorics, number theory, Counting, Modular Arithmetic, Casework

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