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Problem 712

AMC 10/12, early questions

Number theory Difficulty 3.9 Multiple choice Olimpiadi di Matematica · Italy

For how many integers $n$ is $\frac{3 n}{n+5}$ an integer and divisible by $4$ ?

Pick one

Official solution

Solution:

The answer is (C). Substituting $m = n + 5$ , the given expression becomes $\frac{3m - 15}{m}$ , that is $3 - \frac{15}{m}$ . For it to be an integer, then, $m$ must be a divisor of $15$ , so the possibilities are only $1, 3, 5, 15$ and their opposites. Among these, the only ones for which the expression is a multiple of $4$ are $1, -3, 5$ and $-15$ , so the required $n$ are $-20, -8, -4$ and $0$ .

Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty, ordering) added by this project.