🔥 New! 2025 AMC 10 B

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Problem 10 Easy

Let f ( n ) = n ^ { 3 } - 5 n ^ { 2 } + 2 n + 8 ,and let g ( n ) = n ^ { 3 } - 6 n ^ { 2 } + 5 n + 1 2. What is the sum of all integer values of n for which \frac { f ( n ) } { g ( n ) } is also an integer?

  • A.

    2

  • B.

    3

  • C.

    4

  • D.

    5

  • E.

    6

Answer:A
Solution

We have the following observations: \frac{f(n)}{g(n)} - 1 = \frac{f(n) - g(n)}{g(n)} \in \mathbb{Z}

f(n) - g(n) = n^2 - 3n - 4 = (n-4)(n+1)

g(n) = (n+1)(n-4)(n-3)

\frac{f(n) - g(n)}{g(n)} = \frac{1}{n-3} when n \neq -1, 4, 3

For \frac{1}{n-3} \in \mathbb{Z}, we need n - 3 = \pm 1, so n = 2 or 4.

However, n \neq 4, thus n = 2.

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