Got it on the memory I suspected you were motivated by the price sometimes that just doesn't work out. 🙂
Here is the Apple Article that states which iMac's are 32 bit and which are 64 bit:
http://support.apple.com/kb/HT3696
It pretty clearly says the Core 2 Duo is a 64 bit processor. However the 24" late 2007 iMacs came with both a Core Duo (2.4 GHz) or the optional Core 2 Duo (2.8 GHz), the Core Duo is 32 bit the Core 2 Duo is 64 bit. Look closely at System Profiler to see which processor you have.
Now to get your iMac to boot in 64 bit all you need to do is hold down the 6 and 4 keys when it boots up.
If you want to know if your iMac has the 64 bit kernel run this command in Terminal:
ioreg -l -p IODeviceTree | grep firmware-abi
The result will say:
EFI64 is it's 64 bit
or EFI32 if it's 32 bit.
That should do it.
Roger