9. (10 points) Given that $2^{n}-1$ is a multiple of 2015, then the smallest positive integer $n$ is

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9. (10 points) Given that $2^{n}-1$ is a multiple of 2015, then the smallest positive integer $n$ is
【Solution】Solution: Since $2015=5 \times 13 \times 31, 2^{4a}-1$ (where $a$ is a positive integer) is a multiple of 5, $2^{5b}-1$ (where $b$ is a positive integer) is a multiple of 31, $2^{12c}-1$ (where $c$ is a positive integer) is a multiple of 31.
The least common multiple of 4, 5, and 12 is 60,
Therefore, $2^{60}-1$ is a multiple of 2015;
Hence, the answer is 60.