Divisor–Window Formula for the Euclidean Decomposition of the Multiplication Table

This note derives a closed formula for the cells of the Euclidean multiplicity array A(n,q,r) associated with the n X n multiplication table. Apart from the zero-product cell, each entry is exactly the number of divisors of q*n + r lying in the interval (q,n).

The formula gives a direct arithmetic interpretation of the joint quotient–remainder distribution as a moving divisor window across the integers. Its cardinality determines multiplicity, its nonemptiness determines support, and complementary pairing within the window determines parity.

Version V. Further revision and expansion following review.