Question 178
A number bracelet in base $m$ is made by choosing two nonnegative integers less than $m$ (not both $0$) and continuing in a clockwise loop, each succeeding number being the mod $m$ sum of its two predecessors. The figure is closed up as soon as it starts to repeat. The figure to the right shows two number bracelets in base $10$, starting with the pairs $(1,3)$, and $(2,2)$, respectively. Prove that the lengths of all number bracelets in a given base are divisors of the length of the number bracelet beginning with $(0,1)$.