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2000 AMC 12 Problem 16

Problem 16 of 25IntermediateNumber TheoryProblem-Solving Techniques

A checkerboard of 1313 rows and 1717 columns has a number written in each square, beginning in the upper left corner, so that the first row is numbered 1,1, 2,2, …,\ldots, 17,17, the second row 18,18, 19,19, …,\ldots, 34,34, and so on down the board. If the board is renumbered so that the left column, top to bottom, is 1,1, 2,2, …,\ldots, 13,13, the second column 14,14, 15,15, …,\ldots, 2626 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).

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Solution

The square (m,n)(m, n) is numbered 17(m−1)+n17(m - 1) + n originally and 13(n−1)+m13(n - 1) + m after renumbering. Setting these equal gives 4m−3n=1. 4m - 3n = 1. The solutions with 1≤m≤131 \le m \le 13 and 1≤n≤171 \le n \le 17 are (1,1),(1, 1), (4,5),(4, 5), (7,9),(7, 9), (10,13),(10, 13), and (13,17).(13, 17). These squares hold the numbers 1,56,111,166,1, 56, 111, 166, and 221,221, whose sum is 555.555. Thus, the correct answer is D.
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Tagged: Diophantine Equation · systematic listing

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