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December 30th, 2014, 04:18 PM   #1
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Is there infinite many k such that (k+1) | 10^k-1 ?

Suppose $k,\,d\in\mathbb{N}^+$. Prove or disprove:

Set $\{k : (k+1 \mid 10^k -1)\wedge \big(((0 < d< k)\wedge (d \mid k))\implies (k+1\nmid 10^d -1)\big)\}$ is not finite.
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