In this problem we will work with strings over the alphabet $\{$a$,$ b$\},$ and let our $$multipli$-$

cation$$ be string concatenation. We say that one string u $$divides$$ another string v if

$$x : $$y : xuy $=$ v$,$ or equivalently, if u is a substring of v$.$ As with the naturals, we redefine

the word $$prime$$ for strings so that P $($w$)$ means $$w$=$ and if D$($x$,$ w$)$ is true, then either

x $=$ or x $=$ w