again I don't know which of the questions are to be answered.

All numbers here are integers, they differ only in literal representation - i.e. binary, octal, hex or decimal.

Internally they are always only the same bits, i.e. you can mix representations in binary operations.

07777 stands for 4 groups with 3 bits (2**3 =8 => octal).

the highest group is quite special and include "exotic" things like sticky-bit. The others are for user, group and others with rwx -bits,

But there are more bits set in mode to code the file-type:

from stat

               Because the mode contains both the file type and its
               permissions, you should mask off the file type portion 
+and
               (s)printf using a "%o" if you want to see the real perm
+issions.

                   $mode = (stat($filename))[2];
                   printf "Permissions are %04o\n", $mode & 07777;
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Any questions left? =)