Question

Let $f(n) = n^3 - 5n^2 + 2n + 8,$ and let $g(n) = n^3 - 6n^2 + 5n + 12.$ What is the sum of all integer values of $n$ for which $\dfrac{f(n)}{g(n)}$ is also an integer?

$2$

$3$

$4$

$5$

$6$

Answer: A

Concepts:polynomial factoring divisibility

Difficulty rating: 1510

Solution:

Factor both cubics: $f(n) = (n + 1)(n - 2)(n - 4)$ and $g(n) = (n + 1)(n - 3)(n - 4).$ Away from $n \in \{-1, 3, 4\},$ where $g$ vanishes or the ratio is $\tfrac{0}{0},$ the common factors cancel and $\dfrac{f(n)}{g(n)} = \dfrac{n - 2}{n - 3} = 1 + \dfrac{1}{n - 3}.$ That's an integer only when $n - 3 = \pm 1,$ so $n = 2$ or $n = 4.$ But $n = 4$ kills $g,$ so only $n = 2$ survives, and the sum is $2.$ Therefore, the answer is A.

2025 AMC 10B Problem 10

Answer: A

Difficulty rating: 1510

Solution:

Problem 10 in Other Years