SOLUTION: The product of three consecutive integers is 185136. What is the sum of these integers?

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Question 1046508: The product of three consecutive integers is 185136. What is the sum of these integers?
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
let three consecutive integers be +x,%28x%2B1%29,and%28x%2B2%29
given: The product of three consecutive integers is 185136.
so, we have
+x%28x%2B1%29%28x%2B2%29=185136 ...solve for x
+%28x%5E2%2Bx%29%28x%2B2%29=185136
+x%5E3%2Bx%5E2%2B2x%5E2%2B2x=185136
+x%5E3%2B3x%5E2%2B2x-185136=0...factor
+x%5E3%2B59x%5E2-56x%5E2%2B3306x-56%2A59x-56%2A3306=0
+%28x%5E3-56x%5E2%29%2B%2859x%5E2-56%2A59x%29%2B%283306x-56%2A3306%29=0
+x%5E2%28x-56%29%2B59x%28x-56%29%2B3306%28x-56%29=0
%28x-56%29%28x%5E2%2B59x%2B3306%29+=+0
real solution is:
if%28x-56%29=+0->highlight%28x=56%29
%28x%5E2%2B59x%2B3306%29+=+0...coefficient c=3306 is positive and greater than b=59 and we will have complex solutions; so, disregard it

first integer is highlight%2856%29
next one is highlight%2857%29
next one is highlight%2858%29
check their product: 56%2A57%2A58=185136 which is true
now find their sum: 56%2B57%2B58=171

so, the sum of these integers is highlight%28171%29

Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
.
The product of three consecutive integers is 185136. What is the sum of these integers?
~~~~~~~~~~~~~~~~~~~~~~

Take the cube root of this number: root%283%2C185136%29 = 56.994.

After that it is clear that your number should be 56*57*58.

All that you need to do now is to check the correctness.

Plus one more step: add the three factors.