2000 AMC 12 Problem 7

Attempt Problem 7 of the 2000 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 2000 AMC 12 solutions, or check the answer key.

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

How many positive integers

b

have the property that

\log_b 729

is a positive integer?

0

1

2

3

4

\log_b 729 = n,

then

b^n = 729 = 3^6

For

b

to be an integer,

n

must be a positive divisor of

6

\log_b 729 = n,

then

b^n = 729 = 3^6,

n

must be a positive divisor of

6.

The possibilities

n = 1, 2, 3, 6

give

b = 3^6, 3^3, 3^2, 3^1,

that is,

729, 27, 9,

and

3.

There are

4

such values of

b.

Thus, the correct answer is E.

Problem 7 in other years