If the product of three consecutive integers is $120$, then the sum of the integers is

### Concept & Strategy Attempting to solve the equation $x(x+1)(x+2) = 120$ results in a cubic equation ($x^3 + 3x^2 + 2x - 120 = 0$), which is practically impossible to solve quickly by formula. The fastest mathematical route is prime factorization or cube root estimation to deduce the integer sequence. ### Step-by-Step Solution * **Given:** The product of three consecutive integers is $120$. * **Method 1: Factorization** * Find the prime factors of $120$: $120 = 2 \times 2 \times 2 \times 3 \times 5$. * Group these prime factors to form three consecutive numbers. We already have a $5$. Let's see if we can form $4$ and $6$. * Yes, $2 \times 2 = 4$. And $2 \times 3 = 6$. * The sequence is $4 \times 5 \times 6 = 120$. * **Method 2: Cube Root Estimation** * Since the numbers are close to each other, their product is roughly the middle number cubed. * $x^3 \approx 120$. * We know $4^3 = 64$ and $5^3 = 125$. The numbers must be centered around $4$ or $5$. * Testing $4, 5, 6$: $4 \times 5 \times 6 = 120$. (Matches perfectly). * The three integers are $4, 5, \text{ and } 6$. * The question asks for their sum: $$\text{Sum} = 4 + 5 + 6 = 15$$ ### Exam Strategy & Shortcut Use the options backward. The options represent the *sum* of the three numbers. Since the numbers are consecutive, the middle number is simply the Sum divided by $3$. * Check (a): Sum $= 9 \Rightarrow$ Middle $= 3$. Sequence: $2, 3, 4$. Product $= 24$ (No) * Check (b): Sum $= 12 \Rightarrow$ Middle $= 4$. Sequence: $3, 4, 5$. Product $= 60$ (No) * Check (d): Sum $= 15 \Rightarrow$ Middle $= 5$. Sequence: $4, 5, 6$. Product $= 120$ (Yes!) This reverse-engineering technique takes seconds and prevents any complex math. ### Common Pitfall The main pitfall is wasting time trying to formally construct and solve a cubic equation algebraically. Aptitude questions are designed to test logic and pattern recognition, not advanced calculus or complex polynomial factoring. ### Final Answer **Therefore, the correct answer is 15.**