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Question 229063: what are three positive integers whose sum equals their product?1,2,3 1+2=3+3=6
1*2=2*3=6
Found 4 solutions by solver91311, jsmallt9, xtinman61x, richwmiller:
Answer by solver91311(24713)   (Show Source):
Answer by jsmallt9(3758)  (Show Source):
You can put this solution on YOUR website! Anoterh tutor has provided the correct answer but did not show how. Here's one way to figure out how that is the correct answer.
Normally one would use x, x+1 and x+2 for the three consecutive (they are supposed to be consecutive?) positive integers. If we use these the equation looks like:
I'll actually solve this later. But I'm going to start by using different expressions.
Let x = the middle integer
x - 1 = the smallest integer
x + 1 = the largest integer
Now the equation look like:
This equation looks easier because- The 1's cancel on the left
- The multiplication is easier on the right side of this equation (compared to the multiplication on the right side of the earlier equation)
Let's solve . Start by simplifying. the 1's cancel on the left and we'll use the  on the (x-1)(x+1) part (remember multiplication is commutative so we can multiply in any order):
(As you can see, this is a much simpler equation than what we would get using x, x+1 and x+2!)
To solve this we'll get one side equal to zero and factor:
We now have a product which is zero. The Zero Product Property tells us that one of the factors must be zero:
 or  or 
Solving these we get:
or  or 
Remembering that x is the middle integer we have three possible solutions:
-1, 0, 1
-3, -2, -1
1, 2, 3
If the problem did not specify positive integers, all three of these sets would be solutions. But since positive integers were specified we have to reject the first two sets which leaves us with the answer provided by the other tutor.
If we don't notice how using x for the middle integer makes things easier, then we have to solve:
Simplify.
Get one side equal to zero and factor:
The right side will factor with factoring by grouping. Factor out the Greatest Common Factor (GCF) of each pair of terms.
Each group has a common factor of (x+3) which we can factor out:
The second factor will factor as a difference of squares:
Using the Zero Product Property:
 or  or 
Solving each:
 or  or 
Remembering that x is the smallest integer for this equation we get the following sets of 3 consecutive integers:
-3, -2, -1
-1, 0, 1
1, 2, 3
These are the same sets of integers we got earlier. And again we need to reject the first two because the problem specifies positive integers.
Answer by xtinman61x(2) (Show Source):
Answer by richwmiller(17219)  (Show Source):
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