If the average of three different positive integers is 6, how will the product of the three integers be compared with 25?

Question

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.

Answer 1

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.

Answer 2

Assume (x+y+z)/3=6 thus x+y+z =18. The largest of x,y, and x can be at most 15. (15+2+1)=18. This gives us a volume, (xyz), of 30 units cubed. This is also the smallest volume you can get. This can be visualized in 3D space. Every other combination would have three non zero terms. I haven’t worked out a proof of this but I assume it’s trivial although possibly tedious. I don’t know how much help this will be. But you don’t need to test all the other possibilities.