Question 98
In a bazaar, there is a carpet-changer. If he gets from a client an $a \times b$ carpet, he can give instead either a $\frac{1}{a} \times \frac{1}{b}$ carpet, or two carpets of sizes $c \times b$ and $\frac{a}{c} \times b$ (at each such change, a number $c$ is chosen by the client). A traveler tells that initially he had one carpet whose side lengths were greater than $1$, and after a series of changes each of his carpets had a side longer than $1$ as well as a side shorter than $1$. Could this be a correct story? (At the client's request, the changer is ready to regard an $a \times b$ carpet as a $b \times a$ carpet.) (G. Zhukov)