Question

方阵是一个乘法幻方。也就是说，每一行、每一列和每条对角线上的数的乘积都相同。若所有项都是正整数， $g$ 的可能值之和是多少？ $\begin{array}{|c|c|c|} \hline 50 & b & c \\ \hline d & e & f \\ \hline g & h & 2 \\ \hline \end{array}$

The square $\begin{array}{|c|c|c|} \hline 50 & b & c \\ \hline d & e & f \\ \hline g & h & 2 \\ \hline \end{array}$ is a multiplicative magic square. That is, the product of the numbers in each row, column, and diagonal is the same. If all the entries are positive integers, what is the sum of the possible values of $g?$

$10$

$25$

$35$

$62$

$136$

答案：C

知识点：幻方方程组整除性

难度评级：1940

解答：

由各行、列、对角线乘积相等，可把每一项都写成关于 $b$ 的式子： $h = \dfrac{100}{b}$ 、 $g = \dfrac{100}{c}$ 、 $f = \dfrac{100}{d}$ 。比较行和列得 $c = \dfrac{20}{b}$ 、 $d = \dfrac{4}{b}$ ，因此 $g = 5b$ ，且 $e = 10$ 。

所有项都是正整数当且仅当 $b = 1, 2$ ，或 $4$ ，对应 $g = 5, 10, 20$ 。它们的和为 $35$ 。

因此正确答案是 C。

From the equal row, column, and diagonal products, every entry can be written in terms of $b:$ $h = \dfrac{100}{b},$ $g = \dfrac{100}{c},$ $f = \dfrac{100}{d}.$ Comparing rows and columns gives $c = \dfrac{20}{b}$ and $d = \dfrac{4}{b},$ hence $g = 5b$ and $e = 10.$

All entries are positive integers exactly when $b = 1, 2,$ or $4,$ giving $g = 5, 10, 20.$ Their sum is $35.$

Thus, the correct answer is C.

2004 AMC 12B 第 22 题

答案：C

难度评级：1940

解答：

其他年份的第 22 题