MS keeps a hardware "fingerprint" of installations that have "digital entitlement" to Win10 (derived from upgrading from a valid Win7/8 installation) and uses this to "authenticate" a re-installation. All you need is the installation media for the right version, language & bitness of Win10. I've reinstalled Win10 on laptops that had been upgraded from Win7, then wiped.
Go here using a non-Windows computer to download an ISO. If you use a Windows computer it downloads a tool that gets rather annoying.
https://www.microsoft.com/en-au/soft...load/windows10