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

2014 AMC 12B Problem 7

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For how many positive integers

n

\dfrac{n}{30-n}

also a positive integer?

4

5

6

7

8

Rewrite

\dfrac{n}{30-n} = \dfrac{30}{30-n} - 1

30-n

must be a positive divisor of

30,

and the quotient must be at least

2

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.