2004 AMC 12B Problem 22

Question

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

Answer: C

Concepts:magic square system of equations divisibility

Difficulty rating: 1940

Small hint:

Express every entry in terms of

b

using the equal products

Big hint:

The corner relations force

c = \dfrac{20}{b}

and

d = \dfrac{4}{b},

so

b \in \{1, 2, 4\}

Solution:

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.