Doesn't look like we can use gcds for the cells in (e',1) where the a'[t] values are our a multiplied by something. There are lots of gcds greater than 1. Our a isn't even the highest prime gcd out of all of them. This one, for example, has a gcd of 73 between the a values in (390,1,32) and (390,1,43), which is larger than a=43.
Chris, I really want to know, why do you keep telling us you're going to explain how part of this works only to revert back to giving us hints every time? I mean if you want us to figure it out rather than being explicitly told then why would you even tell us you're going to explain it on a particular date? It's very discouraging.