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Almost all 32-bit software should work fine on an 64-bit system

There are problems with device drivers and low-level software, but something like Premiere shouldn't cause any problems at all.
I'm guessing you are using the 64 bit version of Internet Explorer. This does have problems with Adobe Flash, as that is written in 32-bit code and won't work with the 64-bit container application. This is probably the only compatibility problem you will have with bit versions, and ond all other x64 Windows users have at the moment. You'll need to use the default 32-bit version of Internet Explorer to use Adobe Flash. There's no noticeable difference between the two.

Just click on "Internet Explorer" rather than "Internet Explorer (64-bit)" instead, and then download Flash.