Problem: (1970 IMO – #1) Find all positive integers  such that the set  can be partitioned into two subsets so that the product of the numbers in each subset is equal.
Solution: We claim that no such exists.
First, suppose one of the elements has a prime factor  . Then clearly none of the other elements has the same prime factor; therefore, any partition will create one set with that prime factor and one without it. Hence none of the elements has a prime factor .
Now take the set modulo  . By the same argument above, there must be two elements divisible by . But the first and last are the only two elements that differ by , so  .
Consider the elements  . Two of them can be divisible by  . Of the remaining two, however, only one can be divisible by  (since they differ by ). Hence there exists an element that is divisible by neither  nor  . Then it must have a prime factor , which gives us a contradiction.
So no such  exist, as desired. QED.
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Practice Problem: Find an such that the above condition holds for the set  or prove that none exist.
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