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

In a previous note the joint distribution arising from the Euclidean decomposition of the multiplication table was introduced. For 0 <= i,j < n every product admits a representation ij = qn + r with 0 <= q,r < n. The quantity A(n,q,r) = #{(i,j) : 0 <= i,j < n, ij = qn + r} counts the multiplicities of quotient-remainder pairs occurring in the table. In this short note we derive a closed formula for these cell counts. For qn + r > 0 each entry equals the number of divisors of qn + r lying in the interval (q,n). Consequently the array A(n,q,r) admits a natural interpretation as a moving divisor window across the integers qn + r. This result provides a direct arithmetic interpretation of the joint quotient-remainder distribution introduced in the note "Euclidean Decomposition of the Multiplication Table" (2026). Version 2. The title has been revised for consistency with the note itself.