如图所示,连接$ OB $.
$ \because AC $是$ \odot O $的内接正六边形的一边.
$ \therefore \angle AOC = \frac{360°}{6} = 60° $.
$ \because BC $是$ \odot O $的内接正八边形的一边,
$ \therefore \angle BOC = \frac{360°}{8} = 45° $,
$ \therefore \angle AOB = \angle AOC - \angle BOC = 60° - 45° = 15° $.
$ \because \frac{360°}{15°} = 24 $, $ \therefore n = 24 $,
$ \therefore AB $为$ \odot O $的内接正二十四边形的一边.