Problem:
For some positive integer k, the repeating bas-k representation of the (base-ten) fraction 517 is 0.23k=0.23232323⋯k. What is k?
Answer Choices:
A. 13
B. 14
C. 15
D. 16
E. 17
Solution:
The number 0.23k is the sum of an infinite geometric series with first term k2+k23 and common ratio k21. Therefore the sum is
1−k21k2+k23=k2−12k+3=517
Then 0=7k2−102k−160=(k−16)(7k+10), and therefore k=(D)16.
OR
Let x=0.23k. Then (k2−1)x=23.0k=2k+3, so k2−12k+3=517 and the solution proceeds as above.
Note: If 0<a<q,gcd(a,q)=1, and gcd(k,q)=1, then the base- k representation of the fraction qa has least period equal to the order of k modulo q. In the case at hand, k=16 and q=51=3⋅17. Then ϕ(q)=2⋅16=32 and k2=256=5⋅51+1, so k has order 2 modulo 51.
The problems on this page are the property of the MAA's American Mathematics Competitions