The multiple of 12 observation seems to be a robust conclusion for a of 5 or above. Between the 1cc and aabb cases, the difference in x works out algebraically to a^2-1, or (a+1)(a-1).
Since a is odd, because it's prime, both a+1 and a-1 are even, for two factors of 2. And one of a-1, a, or a+1 has to be a multiple of 3, and it can't be a because again, it's prime.
So a^2-1 has to be a multiple of 223 = 12. Neat.