Question 249142: The product of three consecutive integers is 2184 . What is the sum of the integers ?
(A)34 (B) 33 (C) 36 (D) 39 (E) none of these
Answer by chosenpoint(26)   (Show Source):
You can put this solution on YOUR website! A standard word problem involving consecutive integers.
Easy Trial and Error Solution (which would be the way to go on a test, scroll down for the hard solution!):
Now, since we are given possible choices, we can just try trial and error to solve this problem and skip the entire cubic equation we will see in the hard solution below this one.
For example, I can just try different numbers on my calculator to see what I come up with. I know that 10*10*10 is 1000, so I will start with 10 as my first integer:
10*11*12=1320 nope, too low so I will go to 11
11*12*13=1716 again, too low, so I will go to 12
12*13*14=2184 bingo!
then to get the answer, I just add the three integers together
12+13+14=39 final answer D
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Alternative Hard Mathematical Solution (scroll up for the easy solution!):
Let x be the first integer.
Then the three integers are:
x, x+1, and x+2
To solve, we just use the given info about their products:
 given
 distributing
 simplifying
 distributing
 subtracting 2184 from both sides
Now we can use a property of cubic equations.
IF the equation has an integer root, then the integer root MUST be a factor of 2184, the constant part of the equation.
So let's factor 2184:
Doing a quick prime factorization, the factors of 2184 that can be a possible root are ALL the positive/negative factor values (all the possible combinations of the prime factors):
1, 2, 4, 6, 7, 8, 12, 13, 21, 28, 36...2184
since 
So the integer root we are looking for must be somewhere on that list above!
Doing quick math, we can now see that 12*13*14=2184
Since we have verified that our x=12, now we just need to sum up the three integers.
12+13+14=39 final answer is 39 D
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