13. A motorbike and a bicycle travel a large square track, starting at the same time from one of the vertices and both proceeding clockwise. The side of the track measures $90 \mathrm{~km}$. The motorbike travels at a constant speed of $65 \mathrm{~km}$ per hour, the bicycle at $30 \mathrm{~km}$ per hour. After how many hours will the two meet again at one of the four vertices of the track?
(A) 7
(B) $72 / 7$
(C) $30 / 7$
(D) 72
(E) it will never happen again
Problem 13. The answer is (D).

Let $t$ be the time elapsed, measured in hours. The distance traveled by the motorbike is $65 t$ kilometers, and that traveled by the bicycle is $30 t$ kilometers. The problem requires finding the minimum positive value of $t$ for which $30 t$ is a multiple of 90 and the difference in kilometers traveled, $65 t - 30 t = 35 t$, is a multiple of 360. Setting $30 t = 90 m$ and $35 t = 360 n$, with $m, n$ being positive integers, we get $t = 3 m$ and thus, substituting into the second equation and simplifying, $7 m = 24 n$. Therefore, $m$ is a multiple of 24, and the time $t$ will be a multiple of 72. It is then straightforward to verify that all multiples of 72 are solutions, and thus the minimum positive value for $t$ is 72.

(Problem proposed by A. Bianchi)