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).

7.

How many positive integers bb have the property that logb729\log_b 729 is a positive integer?

00

11

22

33

44

Answer: E
Concepts:logarithmfactor countingperfect power
Difficulty rating: 1370
Solution:

If logb729=n,\log_b 729 = n, then bn=729=36,b^n = 729 = 3^6, so nn must be a positive divisor of 6.6.

The possibilities n=1,2,3,6n = 1, 2, 3, 6 give b=36,33,32,31,b = 3^6, 3^3, 3^2, 3^1, that is, 729,27,9,729, 27, 9, and 3.3.

There are 44 such values of b.b.

Thus, the correct answer is E.

← Problem 6#6
Full Exam

Problem 7 in Other Years