I am having trouble with the regularity of the two following languages:

i) $\{0^{n}1^{m}|n,m>0,n-m=0\,mod\,3\}$

ii) $\{0^{n}1^{m}|n,m>0,n+m=0\,mod\,3\}$

To clarify this is stating that the (#(0's) - #(1's)) mod 3 = 0. 

I know how to prove that the first language is regular *without* the n,m > 0 restriction but with it I am inclined to think that the language **is not** regular because we'd need to make sure there is at least one 1 and 0 b and in doing so would also have to "remember" the previous numbers of 0's and/or 1's. Which is not possible with a DFA. 

My logic is similar with the second statement. 

Is my logic right?