L={ a^n : n mod 3 = 0} U { a^n : n mod 5 = 1}
Given:
step 1: n mod 3 = 0
put n = 0 ==> 0 mod 3 = 0
==>a^0 = ∈
step 2 : n mod 5 = 1
put n = 2 ==> 2 mod 5 = 1
==>a^2="aa"
L=(a^0) U (a^2) ==>(∈) U ("aa")
so that "aa" is the minimum string accept by the language and the DFA is given below