Four Digit Octal Modes Against Three Digit Ones

Every octal mode is really four digits; the leading one is just usually zero and usually omitted. It carries the three special bits, each with its own value: setuid is 4, setgid is 2 and sticky is 1. They add, so 6755 is setuid and setgid together, and 7777 is a mode you should never see outside a demonstration.

For a regular file, 755 and 0755 do the same thing. The difference appears on directories under GNU chmod, which deliberately preserves an existing setuid or setgid bit when you give a three digit mode, so that a routine chmod 755 does not silently break the group inheritance on a shared directory. Writing 0755 states the special bits explicitly and clears them.

That behaviour is worth knowing before you run a recursive chmod across a tree that contains setgid directories. A pass of chmod -R 775 leaves the inheritance intact on GNU systems; a pass of chmod -R 0775 removes it, and the symptom appears later as files arriving with the wrong group again. Other implementations do not all make the same choice, so on a mixed estate be explicit about what you want.

Reading a four digit mode is the reverse of writing one. 4755 splits into 4 and 755: setuid, plus owner read write execute, plus read and execute for group and others. 2770 is setgid with a private shared directory. 1733 is a sticky drop box. The leading digit never affects who can read or write, only how the identity or the deletion rule changes.

The calculator always shows both forms side by side, which is the quickest way to check what a leading digit is doing. Set the three special bit checkboxes and watch the leading digit move through 1, 2, 3, 4, 5, 6 and 7 while the last three digits stay put, and the arithmetic stops being something you have to remember.