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We introduce a novel technique that combines the AIM algorithm with the characteristic basis function method (CBFM) to solve the problem of electromagnetic scattering by large but finite periodic arrays. An important advantage of using the CBFM for this problem is that we only need to analyze a single unit cell to construct the characteristic basis functions (CBFs) for the entire array. The CBFs are generated by illuminating a single unit cell with a plane wave incident from different angles, for both the θ- and φ-polarizations. The initial set of CBFs, generated in the manner described above, are then downselected by applying a singular value decomposition (SVD) procedure and retaining only the left singular vectors whose corresponding singular values fall above a threshold. Next, in the conventional CBFM, we derive a reduced matrix by applying the Galerkin procedure and solve it directly if its size is manageable. However, when solving an array problem, which precludes the direct-solve option, we can utilize the adaptive integral method (AIM) algorithm, detailed below, not only to accelerate the solution but to reduce memory requirements as well. Numerical examples are included in this communication to demonstrate the accuracy and the numerical efficiency of the proposed technique.