Problem:
For some positive integer k, the repeating bas-k representation of the (base-ten) fraction 517β is 0.23kβ=0.23232323β―kβ. What is k?
Answer Choices:
A. 13
B. 14
C. 15
D. 16
E. 17
Solution:
The number 0.23kβ is the sum of an infinite geometric series with first term k2β+k23β and common ratio k21β. Therefore the sum is
1βk21βk2β+k23ββ=k2β12k+3β=517β
Then 0=7k2β102kβ160=(kβ16)(7k+10), and therefore k=(D)16β.
OR
Let x=0.23kβ. Then (k2β1)x=23.0kβ=2k+3, so k2β12k+3β=517β 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 qaβ has least period equal to the order of k modulo q. In the case at hand, k=16 and q=51=3β
17. Then Ο(q)=2β
16=32 and k2=256=5β
51+1, so k has order 2 modulo 51.
The problems on this page are the property of the MAA's American Mathematics Competitions