AMC12 2000 A

AMC12 2000 A · Q16

AMC12 2000 A · Q16. It mainly tests Linear equations, Remainders & modular arithmetic.

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).

一个有$13$行$17$列的棋盘，每个格子中写有一个数字，从左上角开始编号，使第一行编号为$1,2,\ldots,17$，第二行编号为$18,19,\ldots,34$，依此类推向下编号。如果将棋盘重新编号，使得最左列从上到下为$1,2,\ldots,13,$，第二列为$14,15,\ldots,26$，依此类推向右编号，则有些格子在两种编号系统中数字相同。求这些格子中的数字之和（在任一系统下相同）。

(A) 222 222

(B) 333 333

(D) 555 555

(E) 666 666

Answer

Correct choice: (D)

正确答案：（D）

Solution

Index the rows with $i = 1, 2, 3, ..., 13$ Index the columns with $j = 1, 2, 3, ..., 17$ For the first row number the cells $1, 2, 3, ..., 17$ For the second, $18, 19, ..., 34$ and so on So the number in row = $i$ and column = $j$ is $f(i, j) = 17(i-1) + j = 17i + j - 17$ Similarly, numbering the same cells columnwise we find the number in row = $i$ and column = $j$ is $g(i, j) = i + 13j - 13$ So we need to solve $f(i, j) = g(i, j)$ $17i + j - 17 = i + 13j - 13$ $16i = 4 + 12j$ $4i = 1 + 3j$ $i = (1 + 3j)/4$ We get $(i, j) = (1, 1), f(i, j) = g(i, j) = 1$ $(i, j) = (4, 5), f(i, j) = g(i, j) = 56$ $(i, j) = (7, 9), f(i, j) = g(i, j) = 111$ $(i, j) = (10, 13), f(i, j) = g(i, j) = 166$ $(i, j) = (13, 17), f(i, j) = g(i, j) = 221$ $\boxed{D}$ $555$

用$i = 1, 2, 3, ..., 13$给行编号，用$j = 1, 2, 3, ..., 17$给列编号。按行编号时，第1行是$1, 2, 3, ..., 17$，第2行是$18, 19, ..., 34$，依此类推。因此第$i$行第$j$列的数字为 $f(i, j) = 17(i-1) + j = 17i + j - 17$。同理，按列编号时，第$i$行第$j$列的数字为 $g(i, j) = i + 13j - 13$。需要解 $f(i, j) = g(i, j)$。 $17i + j - 17 = i + 13j - 13$ $16i = 4 + 12j$ $4i = 1 + 3j$ $i = (1 + 3j)/4$。得到 $(i, j) = (1, 1), f(i, j) = g(i, j) = 1$ $(i, j) = (4, 5), f(i, j) = g(i, j) = 56$ $(i, j) = (7, 9), f(i, j) = g(i, j) = 111$ $(i, j) = (10, 13), f(i, j) = g(i, j) = 166$ $(i, j) = (13, 17), f(i, j) = g(i, j) = 221$ $\boxed{D}$ $555$

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