2024 AMC 10B Problem 25

Each of 27 bricks (right rectangular prisms) has dimensions a \times b \times c, where a, b, and c are pairwise relatively prime positive integers. These bricks are arranged to form a 3 \times 3 \times 3 block, as shown on the left below. A 28th brick with the same dimensions is introduced, and these bricks are reconfigured into a 2 \times 2 \times 7 block, shown on the right. The new block is 1 unit taller, 1 unit wider, and 1 unit deeper than the old one. What is a + b + c?

\textbf{(A)} 88 \qquad \textbf{(B)} 89 \qquad \textbf{(C)} 90 \qquad \textbf{(D)} 91 \qquad \textbf{(E)} 92

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.