Maths Olympiad Prep

Track / Stage 3 / 96 of 260 #96 of 1964

Problem 96

AMC 10/12, early questions
Number theory Difficulty 3.4 Multiple choice

For some positive integer kk, the repeating base-kk representation of the (base-ten) fraction 751\frac{7}{51} is 0.23‾k=0.232323...k0.\overline{23}_k = 0.232323..._k. What is kk?

Pick one

Official solution

We can expand the fraction 0.23‾k0.\overline{23}_k as follows: 0.23‾k=2⋅k−1+3⋅k−2+2⋅k−3+3⋅k−4+⋯0.\overline{23}_k = 2\cdot k^{-1} + 3 \cdot k^{-2} + 2 \cdot k^{-3} + 3 \cdot k^{-4} + \cdots
Notice that this is equivalent to
2(k−1+k−3+k−5+...)+3(k−2+k−4+k−6+⋯ )2( k^{-1} + k^{-3} + k^{-5} + ... ) + 3 (k^{-2} + k^{-4} + k^{-6} + \cdots )
By summing the geometric series and simplifying, we have 2k+3k2−1=751\frac{2k+3}{k^2-1} = \frac{7}{51}. Solving this quadratic equation (or simply testing the answer choices) yields the answer k=(D) 16k = \boxed{\textbf{(D) }16}.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.