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2023 WSMO Team Round Problems/Problem 11

Revision as of 15:37, 2 May 2025 by Pinkpig (talk | contribs) (Created page with "==Problem== When <math>\frac{1}{7}</math> is expressed in base <math>k,</math> the digits are <math>0.a_{k,1}a_{k,2}\dots,</math> where <math>a_{k,1}a_{k,2},\dots</math> are...")
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Problem

When $\frac{1}{7}$ is expressed in base $k,$ the digits are $0.a_{k,1}a_{k,2}\dots,$ where $a_{k,1}a_{k,2},\dots$ are decimal digits. Let $f(p)$ denote the minimum positive integer $x\geq2$ such that $a_{p,1} = a_{p,x},$ for $k\not\equiv0\pmod{7},$ and $1$ for $k\equiv0\pmod{7}.$ Find $\sum_{i=2}^{100}f(i).$

Solution

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