As long as the Gillette Razor has the blades in a cartridge that cannot easily be taken out of the razor then yes the Gillette Razor will be allowed on the airplane in your carry on luggage.
If the Gillette Razor you have has the blades where they're easily taken out of the razor then it will not be permitted on carry on luggage.
You can however take the razor with the blades that come out easily in your checked luggage.