The product of three consecutive positive integers is 8 times the sum of the three numbers. What is the sum of the three integers?

Answer

To solve the problem step by step, we will follow the reasoning laid out in the transcript while providing a clear explanation for each step.

### Step 1: Define the Consecutive Integers
Let the three consecutive positive integers be represented as:
- \( a \) (the first integer)
- \( a + 1 \) (the second integer)
- \( a + 2 \) (the third integer)

### Step 2: Set Up the Equation
According to the problem, the product of these three integers is equal to 8 times their sum. We can express this mathematically as:
\[ a(a + 1)(a + 2) = 8(a + (a + 1) + (a + 2)) \]

### Step 3: Simplify the Right Side
First, simplify the sum on the right side:
\[ a + (a + 1) + (a + 2) = 3a + 3 \]
Thus, the equation becomes:
\[ a(a + 1)(a + 2) = 8(3a + 3) \]

### Step 4: Expand Both Sides
Now, expand the left side:
\[ a(a + 1)(a + 2) = a(a^2 + 3a + 2) = a^3 + 3a^2 + 2a \]
And the right side:
\[ 8(3a + 3) = 24a + 24 \]
So, we have:
\[ a^3 + 3a^2 + 2a = 24a + 24 \]

### Step 5: Rearrange the Equation
Rearranging the equation gives us:
\[ a^3 + 3a^2 + 2a - 24a - 24 = 0 \]
This simplifies to:
\[ a^3 + 3a^2 - 22a - 24 = 0 \]

### Step 6: Solve the Cubic Equation
To solve this cubic equation, we can try to find rational roots using the Rational Root Theorem or by trial and error. Testing \( a = 4 \):
\[ 4^3 + 3(4^2) - 22(4) - 24 = 64 + 48 - 88 - 24 = 0 \]
Thus, \( a = 4 \) is a root.

### Step 7: Find the Consecutive Integers
Now that we have \( a = 4 \), the three consecutive integers are:
- \( 4 \)
- \( 5 \) (which is \( 4 + 1 \))
- \( 6 \) (which is \( 4 + 2 \))

### Step 8: Calculate the Sum
Now, we calculate the sum of these integers:
\[ 4 + 5 + 6 = 15 \]

### Conclusion
The sum of the three consecutive integers is:
\[ \boxed{15} \]