AIME 2025 II · 第 2 题

Solution 1

Since $n + 2$ is positive, the positive factors of $39$ are $1$, $3$, $13$, and $39$.

Therefore, $n = -1$, $1$, $11$ and $37$.

Since $n$ is positive, $n = 1$, $11$ and $37$.

$1 + 11 + 37 = \boxed{049}$ is the correct answer

～Tonyttian

~ Edited by aoum

Solution 2

We observe that $n+2$ and $n+3$ share no common prime factor, so $n+2$ divides $3(n+3)(n^2+9)$ if and only if $n+2$ divides $3(n^2+9)$.

By dividing $\frac{3(n^2+9)}{n+2}$ either with long division or synthetic division, one obtains $3n-6+\frac{39}{n+2}$. This quantity is an integer if and only if $\frac{39}{n+2}$ is an integer, so $n+2$ must be a factor of 39. As in Solution 1, $n \in \{1,11,37\}$ and the sum is $\boxed{049}$.

~scrabbler94

NOTE: If you don't observe that it divides if and only if $n+2$ divides $3(n^2+9)$ you can still do synthetic division by expanding $3(n+3)(n^2+9)$ into $3n^3+9n^2+27n+81$ and get the same remainder. -unhappyfarmer

Solution 3 (modular arithmetic)

Let's express the right-hand expression in terms of mod $n + 2$.

$3 \equiv 3 \pmod{n + 2}$.

$n + 3 \equiv 1 \pmod{n + 2}$.

$n^2 + 9 \equiv 13 \pmod{n + 2}$ since $n^2 - 4 \equiv 0 \pmod{n + 2}$ with a quotient $n - 2$

This means $39 = (n + 2)k \pmod{n + 2}$ where k is some integer.

Note that $n + 2$ is positive, meaning $n + 2 \geq 3$.

$n + 2$ is one of the factors of 39, so $n + 2 = 3, 13,$ or $39$, so $n = 1, 11,$ or $37$.

The sum of all possible $n$ is $1 + 11 + 37 = \boxed{049}$.

~Sohcahtoa157

Solution 4(Polynomial Remainder Theorem) fast and easy

For $n+2$ to be a divisor, the remainder needs to be divisible by $n+2$. Using polynomial remainder theorem, we can plug in $-2$ to get $39$ as our remainder. $\frac{39}{n+2}$ is an integer only when $n = 1, 11, 37$. Hence, the sum of all possible $n$ is $1 + 11 + 37 = \boxed{049}$.

~AncientKarnivore

Video Solution by Mathletes Corner

https://www.youtube.com/watch?v=WYyDe_VyvvQ

~GP102

Video Solutions 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=98hzWL-cUvk&t=6s