Problem:
The national debt of the United States is on track to reach 5×1013 dollars by 2033. How many digits does this number of dollars have when written as a numeral in base 5 ? (The approximation of log105 as 0.7 is sufficient for this problem.)
Answer Choices:
A. 18
B. 20
C. 22
D. 24
E. 26
Solution:
The number of digits required to write the positive integer n in base b is 1+logbn, rounded down to an integer. Therefore the required value is the floor of
1+log5(5⋅1013)=1+log55+13log510=1+1+13⋅log1051≈2+0.713=20.5…
which is (B)20.
OR
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⋅1013=213⋅514. Performing this calculation gives a remainder of 0 for the first 14 iterations. The following table gives the remaining 6 iterations:
dividend8192163832765132quotient1638327651320remainder232032
Therefore 50,000,000,000,000ten =23,023,200,000,000,000,000five , a numeral with (B)20 digits.
The problems on this page are the property of the MAA's American Mathematics Competitions