## Composites with prime repunit property

### Problem 130

Published on Friday, 27th October 2006, 06:00 pm; Solved by 3528; 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.

You are given that for all primes, *p* > 5, that *p* − 1 is divisible by A(*p*). For example, when *p* = 41, A(41) = 5, and 40 is divisible by 5.

However, there are rare composite values for which this is also true; the first five examples being 91, 259, 451, 481, and 703.

Find the sum of the first twenty-five composite values of *n* for which

GCD(*n*, 10) = 1 and *n* − 1 is divisible by A(*n*).