Decimal to octal

Going via binary is usually faster than dividing by eight. Convert to binary, group the bits into threes from the right, and read each group: the grouping is exact because 8 is 2 cubed. This is the same trick that makes hex easy, with three bits instead of four, and it works for any base that is a power of two.

For permissions the digits line up with the bits directly: 493 is 755, which is 111 101 101: read, write and execute for the owner, read and execute for group and others. A fourth digit in front carries the special bits, 4 for setuid, 2 for setgid and 1 for sticky, which is why /tmp is 1777 and a setgid directory is 2775.

The leading zero is the trap. In C a bare 0755 means octal, and that convention survives in enough places to be dangerous; Python 3 and JavaScript in strict mode both reject 0755 outright and want 0o755. The version that bites hardest is passing a decimal number where octal was expected: chmod given the decimal 755 sets octal 1363, which is the sticky bit plus a permission set nobody intended.