Emily has created a jumping game using a straight row of floor tiles that she has numbered $1,2,3,4, \ldots$. Starting on tile 2 , she jumps along the row, landing on every second tile, and stops on the second last tile in the row. Starting from this tile, she turns and jumps back toward the start, this time landing on every third tile. She stops on tile 1. Finally, she turns again and jumps along the row, landing on every fifth tile. This time, she again stops on the second last tile. The number of tiles in the row could be
(A) 39
(B) 40
(C) 47
(D) 49
(E) 53

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-089.jpg?height=41&width=255&top_left_y=2262&top_left_x=935)
Emily has created a jumping game using a straight row of floor tiles that she has numbered $1,2,3,4, \ldots$. Starting on tile 2 , she jumps along the row, landing on every second tile, and stops on the second last tile in the row. Starting from this tile, she turns and jumps back toward the start, this time landing on every third tile. She stops on tile 1. Finally, she turns again and jumps along the row, landing on every fifth tile. This time, she again stops on the second last tile. The number of tiles in the row could be
(A) 39
(B) 40
(C) 47
(D) 49
(E) 53

## Solution

Since Emily first starts on tile 2 and jumps on every second tile, then she lands only on even numbered tiles. Since she stops on the second last tile, the total number of tiles is odd.

Next, Emily jumps back along the row by 3's and ends on tile 1. So every tile that she lands on this time has a number which is 1 more than a multiple of 3 (eg. 1, 4, 7, etc.) So the second last tile has a number that is 1 more than a multiple of 3 . This tells us that the overall number of tiles in the row is 2 more than a multiple of 3 .

These two conditions tell us that the total number of tiles cannot be 39,40 or 49 .

Lastly, Emily jumps by 5 's along the row starting at 1 . This says each tile that she lands on has a number that is 1 more than a multiple of 5 . By the same reasoning as above, the total number of tiles in the row is 2 more than a multiple of 5 .

Of the two remaining possibilities (47 and 53), the only one that satisfies this last condition is 47 , and so 47 satisfies all 3 of the required conditions.

(Work back through Emily's steps using the fact that she starts with 47 tiles to check that this does work.)

ANswer: (C)