DFA for a language that accepts the addition of multiple of 2 and multiple of 5
All natural numbers are expressible in the form $2x+5y$, except for $1,3$. You can check this directly, of use the solutino of the Frobenius problem for two denominations. Therefore another way to write your language (assuming the alphabet is $\{a\}$) is
$$
\overline{\{a,a^3\}}.
$$
You can construct a DFA for this language using 5 states, that count up to 4 and then saturate.

If you want to implement the constraint $2x+5y$ directly as an NFA (which is more economical for larger values of $2,5$), then you construct the following NFA:


States: $q_0,a_1,b_1,b_2,b_3,b_4$.
Initial state: $q_0$.
Final states: $\{q_0\}$.
Transition function:
$$
\begin{array}{c|cccccc}
q &amp; q_0 &amp; a_1 &amp; b_1 &amp; b_2 &amp; b_3 &amp; b_4 \\\hline
\delta(q,a) &amp; a_1,b_1 &amp; q_0 &amp; b_2 &amp; b_3 &amp; b_4 &amp; q_0
\end{array}
$$


In words, there is a 2-cycle and a 5-cycle sharing a vertex $q_0$, which is also the initial state and the unique final state.