Which permissions should be given to a file that needs to be opened and edited by the file's owner and opened read-only by the file's group?

0751 gives the owner rwx (7), group r-x (5), and others execute-only (1) - this grants the group execute permission and gives others access, neither of which is required.

0466 gives the owner read-only (4) which prevents editing, and incorrectly grants the group and others read and write (6).

0540 gives the owner read and execute (5), but the owner requires write permission (2) to edit the file.

0640 assigns the owner read and write permissions (6 = 4+2), the group read-only permission (4), and no permissions to others (0). This precisely matches the requirement that the owner can open and edit the file while the group can only open it read-only, with no access granted to others.

0444 grants everyone read-only access, so the owner cannot write or edit the file.