finite automata – Confusion in understanding a DFA question

I am new to DFA and I am trying to understand the following question.

The alphabet is Σ = {0, 1}
Construct DFA for L = {0^n: n is either a multiple of 3 or a multiple of 5 }

The above language does not say anything about the 1s.

When we write strings in this Language, we get
L = {є, 000, 00000, 000000, 000000000, …}
There are no ones in this language.

According to the language description, 0010 would not be a valid string. Right?

So, in the DFA that we construct do we need to show 1s? Can I just leave them out?