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If the square root of the product of three distinct positive [#permalink]

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02 Sep 2006, 10:17

This topic is locked. If you want to discuss this question please re-post it in the respective forum.

If the square root of the product of three distinct positive
integers is equal to the largest of the three numbers, what is the
product of the two smaller numbers?

(1) The largest number is 12.
(2) The average (arithmetic mean) of the three numbers is 20/3

(xyz)^0.5 = z (assume z is the largest)
xyz=z^2
z(z-xy)=0 (Grouping terms on one side)
Since z is the largest, it CANNOT BE ZERO SINCE THE NUMBERS ARE POSITIVE AND DISTINCT.
So, z=xy

(1) gives us the value of z, SUFFICIENT
(2) x+y+z=20; xy+x+y=20 , If x and y are 6 and 2, this statement is satisfied, SUFFICIENT. (Does any other combination sum to 20 ? )

If the square root of the product of three distinct positive
integers is equal to the largest of the three numbers, what is the
product of the two smaller numbers?

(1) The largest number is 12.
(2) The average (arithmetic mean) of the three numbers is 20/3

sqrt xyz = z therfore xyz = z^2 from st one if z^2 = 144 therfore xy = 12

st two

sum of three numbers = x+y+z = 20 but from question stem z = xy

WE KNOW THAT ODD/EVEN CAN NEVER GIVE AN INTIGER THUS Y CAN NEVER BE ODD ie : if y is odd 20-y is odd and Y+1 IS EVEN AND THUS

x = 20-y/1+y WOULD BE A FRACTION ( BUT X IS A POSITIVE INTIGER)

THUS Y COULD BE {2,4,8,} ANY VALUE FOR Y > 8 WILL YIELD 20-Y/1+Y<1 ,THUS BLUGGING IN VALUES FOR Y WE SEE THAT ONLY 2 CAN YIELD A POSITIVE INTIGER X = 6

If the square root of the product of three distinct positive integers is equal to the largest of the three numbers, what is the product of the two smaller numbers?

(1) The largest number is 12. (2) The average (arithmetic mean) of the three numbers is 20/3

**EDIT***

(1) #'s are 3,4,12 Then product of least 2#'s=12
#'s are 2,6,12 Then product of least 2#'s=12