2024 AMC 12B Problem 6

Problem 6 of 25EasierAlgebra Number Theory

The national debt of the United States is on track to reach

5 \cdot 10^{13}

dollars by

2033.

How many digits does this number of dollars have when written as a numeral in base

5?

(The approximation of

\log_{10} 5

0.7

is sufficient for this problem.)

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Solution

The number of digits of

N

in base

5

\lfloor \log_5 N \rfloor + 1.

With

N = 5 \cdot 10^{13},

\log_{10} N = 13 + \log_{10} 5 = 13.7.

Converting bases,

\log_5 N

= \dfrac{13.7}{\log_{10} 5}

= \dfrac{13.7}{0.7}

= 19.57\ldots

Thus the number of digits is

\lfloor 19.57 \rfloor + 1 = 19 + 1 = 20.

Thus, the correct answer is B.