If you have lost your product key for windows 10 then you don't need to buy a new one. You can activate your account without any difficulties.
A digital licence (the Windows 10 equivalent of a product key) is stored in the BIOS. When Windows 10 is reinstalled the licence is checked to see that the existing licence matches the Windows 10 edition being installed and windows is automatically activated in the absence of any discrepancy.
As long as Windows 10 has been previously installed and activated any subsequent installation matching the earlier installation should automatically activate.
If you have changed the motherboard and the original installation was an OEM copy of Windows you will need to buy a new copy of Windows. If you have changed one or more other hardware devices the new installation may not automatically activate. In this situation you should telephone Support for assistance using the number on the Activation screen.