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Removing digits preserves n|b - OIMU 2008 Problem 1

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August 28, 2010
number theory proposednumber theory

Problem Statement

Let nn be a positive integer that is not divisible by either 22 or 55. In the decimal expansion of 1n=0.a1a2a3⋯\frac{1}{n}= 0.a_1a_2a_3\cdots a finite number of digits after the decimal point are chosen arbitrarily to be deleted. Clearly the decimal number obtained by this procedure is also rational, so it's equal to ab\frac{a}{b} for some integers a,ba,b. Prove that bb is divisible by nn.
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