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This problem trains for: SAT-I, AMC-8, GMAT.

How many pairs of numbers x (x is an irreducible fraction) and n (n is an integer) are there such that:

equation

is true?

We get five solutions:

equation

equation

equation

equation

equation

and another five, from taking the reciprocals of the fractions and the negative of the powers in the first five:

equation

equation

equation

equation

equation

In total 10 solutions.