Could you elaborate? I’m a music teacher and it checks out to me.

C4 up a 5th is G4, then up a fourth is C5. C4 to C5 is an octave.

Two octaves is a 15th. In music when the composer wants you play up an octave, they write (typically) 8va. When they want you to play up two octaves, they write (typically) 15va.