at a factory

I agree with Ellie about it being nit-picky. It's a regional thing. She and I are both native speakers but I would say "I work in a factory." as would others in my region. I also agree with her that it would be "Do you have an account at this bank." Things like that can be very regional. For instance, here in the Southeast, we say "We are standing in line for tickets to this show." where in New York City they would say "We are standing on this line for tickets to the show." On line or in line, it means the same things.

Really all of these options are right, it's just very nit-picky things about "what sounds right." I would say "I work at a factory." and "Do you have an account at this bank?"