Question

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 $\log_{10} 5$ as $0.7$ is sufficient for this problem.)

$18$

$20$

$22$

$24$

$26$

Answer: B

Concepts:logarithm number base

Difficulty rating: 1370

Solution:

The number of digits of $N$ in base $5$ is $\lfloor \log_5 N \rfloor + 1.$ 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.

2024 AMC 12B Problem 6

Answer: B

Difficulty rating: 1370

Solution:

Problem 6 in Other Years