2014 AMC 12B Problem 7 — Solution | LIVE by Po-Shen Loh

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2014 AMC 12B Problem 7

Below is the professionally curated solution for Problem 7 of the 2014 AMC 12B, 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 of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:divisibility factor algebraic manipulation

Difficulty rating: 1520

7.

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

$4$

$5$

$6$

$7$

$8$

Solution:

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.

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