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On my applications, it seems as if I did not fill in my SSN. Is this a big problem? Are there any remedies I can apply?

I doubt it's a big deal if it wasn't a required part of the application. I tend to not put my SSN on applications and other documents for security reasons, unless it's imperative that they have it. If you are really concerned, just ask the school.

It's probably better not to. Harvard specifically says that they don't want your Social on anything, just your LSAC number.

they don't really need it until you actually attend IMO