Euclidean Decomposition of the Multiplication Table

This note introduces the array A(n,q,r) recording the joint distribution of quotient and remainder obtained by applying Euclidean division to the entries of the n X n multiplication table. Several classical arithmetic quantities appear as projections of this array, including quotient and remainder distributions, divisor counts, multiplicative support, and the set of distinct products occurring in the multiplication table. The normalized array also has a direct probabilistic realization through the product of two independent fair zero-based n-sided dice, while parity is determined exactly by diagonal square contributions.