1992 AMC 12 Problem 10

Attempt Problem 10 of the 1992 AMC 12 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 1992 AMC 12 solutions, or check the answer key.

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

10.

The number of positive integers

k

for which the equation

kx-12=3k

has an integer solution for

x

3

4

5

6

7

Solve the equation for

x

in terms of

k

The expression

3+\frac{12}{k}

is integral exactly when

k

is a positive divisor of

12

Solving gives

x=3+\frac{12}{k}.

Thus

x

is an integer precisely when

k

is a positive divisor of

12.

The possibilities are

1,

2,

3,

4,

6,

and

12,

for a total of

6.

Thus the correct answer is D.

Problem 10 in other years