2000 AMC 12 Problem 7

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Concepts:logarithmfactor countingperfect power

Difficulty rating: 1370

7.

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

00

11

22

33

44

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 7 in Other Years