Core 2 processors have no problem with Windows 7/8 or 10.
Even core 2 duo processors have no problem with running windows smoothly. You do need atleast 2GB's of RAM and preferably 4GB or more.
Core 2 quad processors even allow for some gaming, League of legends, WoW, Far cry 3 and the need for speed series.
If you are not going to do any heavy tasks on it however, you should be just fine. But games that i listed above are all playable along with a decent graphics card.