The Support of the Euclidean Decomposition of the Multiplication Table

The multiplication table of order n is one of the oldest arithmetic objects.One may study either the multiplicity of products or the set of distinct products appearing in the table. Euclidean decomposition refines both by recording, for each product i*j, its quotient and remainder upon division by n. This yields the array A(n,q,r), whose entries count how often each pair (q,r) occurs.The resulting support is exactly the binary indicator of the distinct products appearing in the multiplication table. This note formalizes that support, records its equivalent formulations and elementary consequences, and reveals its geometric organization in square-grid form.