If the average of three different positive integers is 6, how will the product of the three integers be compared with 25? This is a practice question from GRE quantitive reasoning: Given the average of three different positive integers is 6. Quantity A: The product of the three integers Quantity B: 25 The question asks to compare A and B. I got the correct answer which is A > B, but I'm wondering if there's a systematic way to solve it (or questions similar to it) without trying all different possible combinations of three integers.

The inequality of arithmetic and geometric means implies that the product can never be larger than $6^3 = 216$. As a rule of thumb, the further you go away from the equal case, the smaller the product gets. So you only need to check the extreme cases, i.e. $(1,2,15)$ here.