How to show the existence of a binary chain

Question

Suppose A, B are finite binary chains that hold AB = BA (* is the concatenation operator). How can I show there exists a binary chain C such that A, B are of the form CCC...C?

Answer 1

By equidivisibility of a free monoid (i.e. the set of finite strings from some alphabet), there exists a $D$ such that either $A = BD = DB$ or $B = AD= DA$. Except in trivial cases ($A = B$, or one of them is the empty string), this now gives you a strictly shorter string to work with. Induction finishes the job.