2000 AMC 12 Problems/Problem 16

Problem

A checkerboard of $13$ rows and $17$ columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered $1,2,\ldots,17$ , the second row $18,19,\ldots,34$ , and so on down the board. If the board is renumbered so that the left column, top to bottom, is $1,2,\ldots,13,$ , the second column $14,15,\ldots,26$ and so on across the board, some squares have the same numbers in both numbering systems. Find the sum of the numbers in these squares (under either system).

$\text {(A)}\ 222 \qquad \text {(B)}\ 333\qquad \text {(C)}\ 444 \qquad \text {(D)}\ 555 \qquad \text {(E)}\ 666$

Solution

Let $(x,y)$ denote the square in row $x \ge 1$ and column $y \ge 1$ . Under the first ordering this square would have a value of $17(x-1) + y$ . Under the second ordering this square would have a value of $13(y-1) + x$ . Equating, $17x-17 + y = 13y-13+x \Longrightarrow 16x = 12y + 4 \Longrightarrow 4x = 3y + 1$ . The pairs that fit this equation are $(1,1),(4,5),(7,9),(10,13),(13,17)$ ; their corresponding values sum up to $1 + 56 + 111 + 166 + 221 = 555\ \mathrm{(D)}$ .

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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2000 AMC 12 Problems/Problem 16

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