See this wiki article which lists what every version of ArchiCAD can read and write:
http://www.archicadwiki.com/ArchiCAD%20versions
As you can see, you can only go backwards one step at a time. So, if you really have to go from 13 back to 10, you will need copies of 12 and 11 and will have to save 13 --> 12 --> 11 --> 10.
Do note that your library parts cannot (in general) move backwards, so a 13 project that you try to save back as version 10 will require a lot of work. You'll get the basic tool elements, but probably not any objects (windows/doors/etc)... unless you modeled the project with the AC 10 library all along for whatever reason.
Cheers,
Karl
One of the forum moderators
AC 28 USA and earlier • macOS Sequoia 15.2, MacBook Pro M2 Max 12CPU/30GPU cores, 32GB