2

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.

YuiTo Cheng
  • 4,705
kastle
  • 75
  • you could try to solve the optimization problem $max(xy)$ under constraint $x+y = h, x \geq 0, y \geq0 $. Using the Lagrangian you should find that the maximum is for $x = y = \frac 1h$.

    An interesting intuitive proof for integers only: https://math.stackexchange.com/questions/1571526/maximize-product-with-sum-constraint

     – Frostic Aug 12 '19 at 15:58
  • As Klaus pointed out in his answer, the smallest product is $30$ for $1\times2\times15$ because product$(2,3,4=24)$ but mean$(2,3,4\lt6)$, and product $(1,6,11)=66$, etc. – poetasis Aug 12 '19 at 17:15

2 Answers2

3

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.

Klaus
  • 10,578
  • Could you please clarify a bit on "the further you go away from the equal case"? Does it mean that, in general, the product is smallest when integers spread out from each other, such that it has some sort of large standard deviation compared with when all integers are equal? – kastle Aug 12 '19 at 17:07
  • Since $(x+1)(y-1) = xy + y - x -1 < xy$ if $y \leq x$, you can always make a product smaller by adding $1$ to a large number and subtracting $1$ to a small number. By repeating this, you only need to check the extreme case, where you cannot add or subtract anything anymore without violating the assumptions. – Klaus Aug 12 '19 at 17:19
0

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.

Jerry
  • 30