Do I need to have both components to support 64-bit, motherboard and CPU. Or is it just the CPU that matters?
To quote Microsoft (eh),
In general, it helps to understand how a 64-bit system differs from the 32-bit one.
The first couple of pages from this old article are also a good read,
A motherboard usually supports some kind of CPUs and may not support some other kind. So, if a motherboard supports X CPU which happens to be 64-bit then it is supported, but that doesn't mean that it will support Y CPU which is 32-bit.
It really drills down to what models of cpus a motherboard supports regardless of 32-bit or 64-bit. Check the motherboard's manual to know which CPUs are fully supported.