2024 AMC 12B Problem 6

Attempt Problem 6 of the 2024 AMC 12B 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 2024 AMC 12B solutions, or check the answer key.

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

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.)

18

20

22

24

26

The number of base-

5

digits of

N

\lfloor \log_5 N \rfloor + 1

Use

\log_5 N = \dfrac{\log_{10} N}{\log_{10} 5}

with

\log_{10} 5 = 0.7

and

\log_{10}(5\cdot 10^{13}) = 13 + \log_{10} 5

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.

Problem 6 in other years