Competition problem — Number theory

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Number theory Difficulty 4.9 AIME Find the answer

10. (10 points) When the fraction $\frac{1}{2016}$ is converted to a repeating decimal, the repeating block has exactly $\qquad$ digits.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

【Answer】Solution: $\frac{1}{2016}=1 \div 2016=0.00049603174603174 \cdots$ , so, the repeating part is 603174, the repeating part has exactly 6 digits. Therefore, the answer is: 6.

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