2008 iTest Problems/Problem 48
Problem
A repunit is a natural number whose digits are all . For instance,
are the four smallest repunits. How many digits are there in the smallest repunit that is
divisible by ?
Solution
We have . Since
and
are relatively prime, it suffices for
, or
.
The smallest integer such that
where
is
, where
is the Carmichael function. For primes
,
, and so the answer is
.