Yes, you can absolutely bring a Gillette Mach 3 (or any similar cartridge-based razor) on a plane in both your carry-on and checked luggage. TSA and international security rules distinguish between "safe" razors and "dangerous" ones. Cartridge razors like the Mach 3, where the blades are permanently set into a plastic housing, are deemed safe because the blades cannot be easily removed to be used as a weapon. This also applies to disposable razors where the handle and head are a single unit. However, you must be careful with safety razors (the traditional metal ones that take a single flat blade); while the handle is allowed in carry-ons, the individual blades are strictly prohibited and must be packed in checked luggage. Straight razors (the "cut-throat" kind used in barbershops) are also banned from carry-ons. So, for your Mach 3, you have zero restrictions—you can even keep it in your toiletry bag and take it through the X-ray without needing to remove it or declare it to the security officers.