2026 AIME II Problem 15

Find the number of ordered 7-tuples (a_1, a_2, a_3, \ldots, a_7) having the following properties:

- a_k \in \{1,2,3\} for all k.

- a_1+a_2+a_3+a_4+a_5+a_6+a_7 is a multiple of 3.

- a_1a_2 a_4 + a_2a_3a_5 + a_3a_4 a_6 + a_4 a_5 a_7 + a_5 a_6 a_1 + a_6 a_7 a_2 + a_7 a_1 a_3 is a multiple of 3.

Leading zeroes must be inputted, so if your answer is 34, then input 034

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