## Repunit divisibility

### Problem 129

Published on Friday, 27th October 2006, 06:00 pm; Solved by 3867; Difficulty rating: 45%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.