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Carter was smart and hide his documents in walls as insulation in homes he built with Habitat for Humanity
Edit: I’m glad so many watch late night television and you feel the need to let me know that you do so. However, I do not. If it bothers you to see similar jokes being made about the same current events, feel free to give him credit. I don’t need you too tell me you watch Colbert.
Reverse National Treasure plot: they all have docs and have to team up to get them back to the Archives. >

I always thought it would be hilarious to put a bunch of former presidents in a anti-buddy-cop movie where they all hate each other but have to team up to save/do something and become best friends along the way. >

There is one! My Fellow Americans (1996). 
“Ex-presidents Kramer (Jack Lemmon) and Douglas (James Garner) have hated each other for years. But when the current administration of President Haney (Dan Aykroyd) -- Kramer's former vice president -- tries blaming a string of kickbacks from a defense contractor on the two former presidents, they band together. With Haney's goons trying to kill them, they set out across the country to try and clear their reputations. In the process, they see a side of America they never knew before.” >

I love this movie. It was one of those movies your family owns on vhs as a kid and you watched dozens of times. When you try to talk to other people about it they look at you like youre from mars. 
Anyways great film, 8 year old persianator loved it, and I still do. >

Well it's a good thing we aren't Gerald Ford or we'd be in a heap at the bottom of these stairs. >

I loved the bit where they shared their own personal lyrics for "Hail to the Chief". >

I love that part, I think of it whenever I hear that song >

The Carter comments are funny, since he predates the presidential records act, and it's likely that any documents he might have would be automatically declassified by this point anyway.
I feel like they included him just to be nice.