The sum of three consecutive integers is equal to their product. How many such possibilities are there?

Let the three consecutive integers be n-1, n, and n+1. 1. Calculate their sum: Sum = (n-1) + n + (n+1) = 3n 2. Calculate their product: Product = (n-1) * n * (n+1) = n(n^2 - 1) 3. Set the sum equal to the product: 3n = n(n^2 - 1) 4. Solve the equation: Rearrange the equation: n(n^2 - 1) - 3n = 0 n( (n^2 - 1) - 3 ) = 0 n(n^2 - 4) = 0 This equation gives us three possible values for n: a) n = 0 b) n^2 - 4 = 0 => n^2 = 4 => n = +2 or n = -2 5. Determine the sets of integers for each value of n: a) If n = 0: The integers are (0-1), 0, (0+1) = -1, 0, 1. Check: Sum = -1 + 0 + 1 = 0. Product = (-1) * 0 * 1 = 0. (Possibility 1) b) If n = 2: The integers are (2-1), 2, (2+1) = 1, 2, 3. Check: Sum = 1 + 2 + 3 = 6. Product = 1 * 2 * 3 = 6. (Possibility 2) c) If n = -2: The integers are (-2-1), -2, (-2+1) = -3, -2, -1. Check: Sum = -3 + (-2) + (-1) = -6. Product = (-3) * (-2) * (-1) = -6. (Possibility 3) There are three distinct sets of consecutive integers that satisfy the condition. The final answer is C) Only three.