32bit or 64bit doesn't matter as far as dual core goes. It is the same for both.
If you run more the one program at a time then both cores get utilized. Each core can only process one thread at a time, so if you have multicores then it can handle multi-threads; either as multiple programs or a program that has been written to be multi-threaded and can thus utilizes both cores.
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RIP Mom 9/17/55-02/22/08.
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