The text itself makes no claims to be written by Moses.
Solid tradition suggests that Moses wrote these books during the 40 years in the desert, in part recording instructions from God, in part preserving the history of the Jewish people.
The bottom line is that, working only with the best evidence available, the OT authors almost “A-to-Z” must be viewed as anonymous.
The authority of the ancient sources and the integrity of the writers who drew upon them to compile the Hebrew Scriptures are not diminished by this formal anonymity.
Evidently scribes who produced this scroll never doubted the singular unity of the book.