2000 AMC 12 Problem 7

Problem 7 of 25EasierAlgebra Number Theory

How many positive integers

b

have the property that

\log_b 729

is a positive integer?

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Solution

\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.

Concepts: logarithm · factor counting · perfect power

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