AMC12 2004 B

AMC12 2004 B · Q22

AMC12 2004 B · Q22. It mainly tests Fractions, Casework.

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 $g$?

正方形 \[ \begin{array}{|c|c|c|} \hline 50 & b & c\\ \hline d & e & f\\ \hline g & h & 2\\ \hline \end{array} \] 是一个乘法幻方。也就是说，每一行、每一列以及两条对角线上的数的乘积都相同。如果所有位置上的数都是正整数，$g$ 的所有可能取值之和是多少？

(A) 10 10

(B) 25 25

(D) 62 62

(E) 136 136

Answer

Correct choice: (C)

正确答案：（C）

Solution

All the unknown entries can be expressed in terms of $b$. Since $100e = beh = ceg = def$, it follows that $h = \frac{100}{b}, g = \frac{100}{c}$, and $f = \frac{100}{d}$. Comparing rows $1$ and $3$ then gives $50bc = 2 \cdot \frac{100}{b} \cdot \frac{100}{c}$, from which $c = \frac{20}{b}$. Comparing columns $1$ and $3$ gives $50d \cdot \frac{100}{c}= 2c \cdot \frac{100}{d}$, from which $d = \frac{c}{5} = \frac{4}{b}$. Finally, $f = 25b, g = 5b$, and $e = 10$. All the entries are positive integers if and only if $b = 1, 2,$ or $4$. The corresponding values for $g$ are $5, 10,$ and $20$, and their sum is $\boxed{\mathbf{(C)}35}$.

所有未知项都可以用 $b$ 表示。由于 $100e = beh = ceg = def$，可得 $h = \frac{100}{b},\ g = \frac{100}{c}$，以及 $f = \frac{100}{d}$。比较第 $1$ 行与第 $3$ 行得 $50bc = 2 \cdot \frac{100}{b} \cdot \frac{100}{c}$，从而 $c = \frac{20}{b}$。比较第 $1$ 列与第 $3$ 列得 $50d \cdot \frac{100}{c}= 2c \cdot \frac{100}{d}$，从而 $d = \frac{c}{5} = \frac{4}{b}$。最后，$f = 25b,\ g = 5b$，且 $e = 10$。所有项均为正整数当且仅当 $b = 1, 2,$ 或 $4$。对应的 $g$ 值为 $5, 10,$ 和 $20$，其和为 $\boxed{\mathbf{(C)}35}$。

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