For how many positive integers n is (n)/(30-n) also a positive integer?

2014 AMC 12B Problem 7 — Solution

Attempt Problem 7 of the 2014 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2014 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

For how many positive integers $n$ is $\dfrac{n}{30-n}$ also a positive integer?

$4$

$5$

$6$

$7$

$8$

Write $\dfrac{n}{30-n} = \dfrac{30}{30-n} - 1.$

For this to be a positive integer, $30-n$ must be a positive divisor of $30$ with $\dfrac{30}{30-n} \ge 2,$ i.e. $30-n \le 15.$

The divisors of $30$ that are at most $15$ are $1, 2, 3, 5, 6, 10, 15,$ giving $7$ values of $n$ (namely $15, 20, 24, 25, 27, 28, 29$ ).

Thus, the correct answer is D.