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Hello. My name is***** to JustAnswer. I'm reviewing your question now, and will post back with your reply momentarily.
You can get into facebook with that gmail because it is tied to your account and they
want only the name for you to get in your facebook as your facebook has a different
password. It doesnt actually access your gmail.
A couple of things first, if you get locked out of facebook and need to recover?
You wont be able to since you are locked out of gmail.
Add another email account and/or phone number to your facebook security recovery
options now if you dont have them so you dont lose that too.
As for your gmail, did you ever access it on the computer you have successfully
and have you tried the recovery forms?