N = pA + qB with p ≡ 1 (mod q)

This paper studies the linear Diophantine representation system N = pA + qB under the condition p ≡ 1 (mod q). We prove that the minimal coefficient A₀ always equals the generalised digital root of N, and derive a closed formula for the total number of non-negative representations together with the threshold values at which this count increases. Two structural theorems are established: an additivity property linking representations of a, b, and a+b on three simultaneous levels, and a Matryoshka property showing that every step of N's digit-sum reduction chain reappears as a coefficient in its representations. A third result, the Palindrome Block Theorem, completely characterises representability below the classical Frobenius bound through a provable, symmetric block structure. The case (p,q) = (19,9), from which this framework originated, is treated throughout as the primary illustrative instance. All results are verified computationally.