2024 AMC 12B Problems/Problem 6
Contents
Problem 6
The national debt of the United States is on track to reach dollars by . How many digits does this number of dollars have when written as a numeral in base 5? (The approximation of as is sufficient for this problem)
Solution
The number of digits is just . Note that
Hence, our answer is
~tsun26
Solution 2
We see that and . Converting this to base gives us (trust me it doesn't take that long). So the final number in base is with zeroes at the end, which gives us digits. So the answer is .
~sidkris
Solution 3
converted to base 5, divide repeatedly by 5 and keep track of the remainders:
1. , remainder
2. , remainder
3. , remainder
4. , remainder
5. , remainder
6. , remainder
Thus, in base 5 is , which has 6 digits When we multiply by , the multiplication shifts the digits by 14 places to the left, adding 14 zeros. Thus, the total number of digits is:
6 + 14 = .