List the first p rep-units beginning with the smallest rep-unit >= p.  

Divide them all by p.  The p remainders will be in the list 0,1,2, . . . , (p-1).  

If any remainder = 0, we're finished.  

Otherwise, we have p remainders chosen from the remaining (p-1) possibilities so some remainders will be duplicates.  

Choose two rep-units with the same remainder.  Call them x and y.  

Then (x-y) is divisible by p since it has a remainder of 0 after division by p.  But (x-y) = (power of ten) times (rep-unit).  Given that p>5 it can't factor (power of ten) so it must factor (rep-unit).