SOLUTION: If the product of 3 consecutive integers is 120,then the sum of the integers is

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Question 889285: If the product of 3 consecutive integers is 120,then the sum of the integers is
Found 2 solutions by dkppathak, Edwin McCravy:
Answer by dkppathak(439)   (Show Source): You can put this solution on YOUR website!
If the product of 3 consecutive integers is 120,then the sum of the integers
solution
let numbers are
X,(X+1) ,(X+2)
then as per conditions
X(X+1)(X+2)=120 to find X+(X+1)+(X+2)=?
X(X+1)(X+2)=4x5x6
than x=4 or x+1=4 or x+2=6
in each case we will find that
X=4
therefor three consecutive number are X,(X+1) ,(X+2) or 3,4,5
therefor sum will be 4+5+6 =15
answer 15
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
Here is a more advanced way of finding them, using the
Cauchy AM-GM inequality.

Both the arithmetic and the geometric means of any three 
unequal numbers is always between the smallest and largest.

The geometric mean of three numbers is the cube root of
their product.

Therefore the geometric mean of the three consecutive integers,
by calculator is approximately 4.932424149

By the Cauchy inequality, the geometric mean of any number of 
unequal positive numbers is always less than their arithmetic 
mean.  

The arithmetic mean of three positive consecutive integers 
is always the middle one. 

The only three consecutive integers which 4.932424149 could 
be between the smallest and largest of and less than the 
middle one, are the positive integers 4,5, and 6.

Their sum is 4+5+6 = 15

Check: 4󬊆 = 120.

Edwin
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