Problem:
For some positive integer k, the repeating base- k representation of the (base-ten) fraction 517 is 0.23k=0.232323…k. What is k?
Answer Choices:
A. 13
B. 14
C. 15
D. 16
E. 17
Solution:
We can expand the fraction 0.23k as follows: 0.23k=2⋅k−1+3⋅k−2+2⋅k−3+3⋅k−4+⋯ Notice that this is equivalent to
2(k−1+k−3+k−5+…)+3(k−2+k−4+k−6+⋯)
By summing the geometric series and simplifying, we have k2−12k+3=517. Solving this quadratic equation (or simply testing the answer choices) yields the answer k= (D)16 .
OR
Let a=0.2323…k. Therefore, k2a=23.2323…k.
From this, we see that k2a−a=23k, so a=k2−123k=k2−12k+3=517.
Now, similar to in Solution 1, we can either test if 2k+3 is a multiple of 7 with the answer choices, or actually solve the quadratic, so that the answer is (D)16 .
The problems on this page are the property of the MAA's American Mathematics Competitions