Question

The product of two consecutive positive integers is 272. What is the larger of the integers?

• A. 15
• B. 19
• C. 17
• D. 18
• E. 16

Hint:

To solve problems involving consecutive integers, set up an equation and use the quadratic formula if necessary.

Answer 1

The Correct Option is C

Step 1: Define the integers. Let the two consecutive integers be \( n \) and \( n+1 \). Their product is given by: \[ n(n+1) = 272. \]

Step 2: Solve the equation. Expanding the equation: \[ n^2 + n = 272. \] Rearranging it into a quadratic equation: \[ n^2 + n - 272 = 0. \]

Step 3: Solve using the quadratic formula.
The quadratic formula is: \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 1 \), \( b = 1 \), and \( c = -272 \). Substituting the values: \[ n = \frac{-1 \pm \sqrt{1^2 - 4(1)(-272)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 1088}}{2} = \frac{-1 \pm \sqrt{1089}}{2}. \]
Since \( \sqrt{1089} = 33 \), we get: \[ n = \frac{-1 + 33}{2} = 16. \]
Thus, the larger integer is \( 17 \).