You should be more careful when making posts.
You claim you have a '2.4Ghz Core Duo' -- the original Core Duo is 32-bit and thus, if this were true, you'd need the 32-bit version.
However, considering there never was a 2.4Ghz Core Duo iMac, I'm assuming you actually have a Core 2 Duo instead. The Core 2 Duo is 64-bit.