Problem:
The square
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 ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
All the unknown entries can be expressed in terms of . Since ceg def, it follows that , and . Comparing rows 1 and 3 then gives
from which .
Comparing columns 1 and 3 gives
from which .
Finally, , and . All the entries are positive integers if and only if , or 4 . The corresponding values for are 5,10 , and 20 , and their sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions