Question

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).

$222$

$333$

$444$

$555$

$666$

Solution:

The square $(m, n)$ is numbered $17(m - 1) + n$ originally and $13(n - 1) + m$ after renumbering. Setting these equal gives $4m - 3n = 1.$

The solutions with $1 \le m \le 13$ and $1 \le n \le 17$ are $(1, 1),$ $(4, 5),$ $(7, 9),$ $(10, 13),$ and $(13, 17).$

These squares hold the numbers $1, 56, 111, 166,$ and $221,$ whose sum is $555.$

Thus, the correct answer is D.

2000 AMC 12 Problem 16

Solution:

Problem 16 in Other Years