(1) Determine a 2-digit positive integer the product of whose digits is one-half the integer. Now, it won't take you long to find such a number (once you get past the elliptical phrasing), but that's just to whet your appetite. The real challenge begins:I have the solution (or is it "a solution"?). Now I have to see it if it is the only solution. I will update this post as I proceed. Right now I have to get ready for school.
(2) Prove that your answer to (1) is unique, i.e., there is only one solution to the problem.
Comment: We're looking for more than an exhaustive search through all ninety 2-digit numbers or a programmed solution. The key to this and all of the remaining questions is to find an approach to solving a single equation which has 2 or more variables whose domain is the set of positive integers. Students are usually not introduced to solving such equations but they appear frequently on SATs and Math Contests. Because we are looking only for positive integer solutions, a standard algebraic approach must be supplemented with arithmetic concepts and testing of several possibilities. Number theorists refer to these as Diophantine equations.
"This investigation was authored by Dave Marain."
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