means that we have a number x, such that when it is divided by 10 it leaves a remainder of 1. 11, 21, 31, etc. are numbers of this type.

So whereas so does 11. So there is a distinct difference between saying "1" and "1 (mod 10)." As far as the (mod 10) vs. mod 10, as far as I know there is no difference.

-Dan