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

The positive integers a and b have:

What is the difference between the largest possible value of a and the smallest possible value of a?

The largest possible value for one of the numbers is to have the same value as the LCM and the smallest possible value is to have the same value as the GCD.

Therefore,

Example of optimizing computations without calculator: