To construct a minimum state deterministic finite automaton (DFA) accepting the language L = {w | w ∈ {0,1}*, the number of 0s and 1s in w are divisible by 3 and 5, respectively}, we need to consider the divisibility of the number of 0s and 1s by 3 and 5, respectively.

Since the language L requires both the number of 0s and 1s to be divisible by 3 and 5, respectively, we need to design the DFA in such a way that it keeps track of the number of 0s and 1s modulo 3 and 5, respectively.

Let's divide the problem into two parts: tracking the number of 0s and 1s modulo 3 and modulo 5.

Tracking the Number of 0s Modulo 3:
- Let's consider the DFA states based on the remainder of the number of 0s divided by 3.
- We can have a total of 3 states: 0, 1, and 2.
- Initially, the DFA is in state 0, as no 0s have been encountered yet.
- Upon encountering a 0, the DFA remains in the same state, as the remainder is unchanged.
- Upon encountering a 1, the DFA transitions to the next state, as the remainder increases by 1.
- After three transitions, the DFA should return to state 0, as the remainder should be divisible by 3.

Tracking the Number of 1s Modulo 5:
- Let's consider the DFA states based on the remainder of the number of 1s divided by 5.
- We can have a total of 5 states: 0, 1, 2, 3, and 4.
- Initially, the DFA is in state 0, as no 1s have been encountered yet.
- Upon encountering a 1, the DFA transitions to the next state, as the remainder increases by 1.
- After five transitions, the DFA should return to state 0, as the remainder should be divisible by 5.

Combining the DFA for the Number of 0s and 1s:
- Now, we need to combine the two DFAs.
- We can create a new DFA with 3 * 5 = 15 states to represent all possible combinations of the remainders of the number of 0s and 1s modulo 3 and 5, respectively.
- The initial state of the DFA is the combination of state 0 for the number of 0s modulo 3 and state 0 for the number of 1s modulo 5.
- For each input symbol (0 or 1), the DFA transitions to the next state based on the current state and the input symbol.
- After analyzing each input symbol, if the DFA is in the state representing the combination of 0 for the number of 0s modulo 3 and 0 for the number of 1s modulo 5, the input string is accepted.

Therefore, the minimum state deterministic finite automaton accepting the language L requires 15 states.