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 ?
If the power of a prime other than divides , then from it follows that , but then considering the product of the diagonals, but , contradiction. So the only prime factors of are and .
It suffices now to consider the two magic squares comprised of the powers of and of the corresponding terms. These satisfy the normal requirement that the sums of rows, columns, and diagonals are the same, owing to our rules of exponents; additionally, all terms are non-negative.
The powers of :
So , so . Indeed, we have the magic squares
The powers of :
Again, we get . However, if we let , then , which obviously gives us a contradiction, and similarly for . For , we get
In conclusion, can be , and their sum is .
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.
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