11. Caterina is jumping along a row of tiles. Starting from the first, with jumps of 2 tiles at a time (i.e., she jumps to the $3^{rd}$, the $5^{th}$, the $7^{th}$, and so on), she reaches the last one; she turns around and, with jumps of 3 tiles at a time, returns to the first; she turns again and, with jumps of 4 tiles, reaches the last tile again; she turns once more and, with jumps of 5 tiles, returns to the first. Which of the following could be the number of tiles in the row?
(A) 121
(B) 91
(C) 90
(D) 60
(E) 31
(11) The correct answer is $(A)$.

Let $m$ be the number of tiles. From the text, it follows that $m-1$ must be a multiple of 2, 3, 4, and 5. Therefore, $m-1$ must be a multiple of the least common multiple of these numbers, which is 60. Thus, it must be $m-1=60k$, with $k$ a positive integer, i.e., $m=60k+1$. Among the indicated answers, only 121 is of this type.

Question proposed by Carmelo Di Stefano.