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Algebra Difficulty 4.9 AIME Find the answer United States

The national debt of the United States is on track to reach $5 \times 10^{13}$ 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.)

Pick one

Solution

The number of digits required to write the positive integer $n$ in base $b$ is $1 + \log_b n$ , rounded down to an integer. Therefore the required value is the floor of
$1 + \log_5 (5 \cdot 10^{13}) = 1 + \log_5 5 + 13 \log_5 10 = 1 + 1 + 13 \cdot \frac{1}{\log_{10} 5} \approx 2 + \frac{13}{0.7} = 20.5\dots,$
which is 20.

It is possible to convert a positive integer to base 5 by repeatedly dividing by 5 and recording the remainders. This list of remainders in reverse order is the required numeral. Here $5 \cdot 10^{13} = 2^{13} \cdot 5^{14}$ . Performing this calculation gives a remainder of 0 for the first 14 iterations. The following table gives the remaining 6 iterations:

dividend	quotient	remainder
8192	1638	2
1638	327	3
327	65	2
65	13	0
13	2	3
2	0	2

Therefore

50,000,000,000,000_{\text{ten}} = 23,023,200,000,000,000,000_{\text{five}}

, a numeral with 20 digits.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.