There are 32-bit and 64-bit Adobe flash players available for Linux. Which you use depends upon the browser you have installed. On a 64-bit system, you can have a 32-bit Firefox installed, which would require a 32-bit plugin. A 64-bit version of FF or Chrome would require a 64-bit plugin. FWIW, I run a 64-bit Linux OS (RHEL 6.1 clone) and Chrome browser w/ 64-bit Adobe Flash plugin. It works fine for me.