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Problem 129


Problem 129


Repunit divisibility

A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k; for example, R(6) = 111111.

Given that n is a positive integer and GCD(n, 10) = 1, it can be shown that there always exists a value, k, for which R(k) is divisible by n, and let A(n) be the least such value of k; for example, A(7) = 6 and A(41) = 5.

The least value of n for which A(n) first exceeds ten is 17.

Find the least value of n for which A(n) first exceeds one-million.


循环单位数整除性

只包含数字1的数被称为循环单位数,我们定义R(k)是长为k的循环单位数,例如,R(6) = 111111。

如果n是一个整数,且GCD(n, 10) = 1,可以验证总存在k使得R(k)能够被n整除,并且记A(n)是这些k中最小的一个。例如,A(7) = 6,而A(41) = 5。

使得A(n)第一次超过十的n是17。

求使得A(n)第一次超过一千万的n。