2004 AMC 12B Problems/Problem 22
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 ?
All the unknown entries can be expressed in terms of . Since , it follows that , and . Comparing rows and then gives , from which . Comparing columns and gives , from which . Finally, , and . All the entries are positive integers if and only if or . The corresponding values for are and , and their sum is .
Credit to Solution B goes to http://billingswest.billings.k12.mt.us/math/AMC%201012/AMC%2012%20work%20sheets/2004%20AMC%2012B%20ws-15.pdf, a page with a play-by-play explanation of the solutions to this test's problems.
We know because this is a multiplicative magic square that each of the following are equal to each other:
From this we know that , thus . Thus and . Thus From this we know that . Thus . Now we know from the very beginning that or or or . Rearranging the equation we have or due to and both being positive. Now that we find all pairs of positive integers that multiply to . There is . Now we know that and b has to be a positive integer. Thus can only be , , or . Thus can only be ,,or . Thus sum of = . The answer is .
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