Can someone help me for better understanding.

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Write the binary values of first 5 multiples of 3.

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Consider the states as the reminders obtained while dividing with 3 ( q0 represents 0 remainder, q1 → remainder 1 and q2 → remainder 2), and transit the binary values of first 4-5 multiples according to remainders the number would give when it would be divided by 3.

Answer 1

It's simple consider the states as remainders when some binary number is divided by 3.i.e.y mod  3==0.Thats why q0 is final state as the remainder will be 0 there only.

For example if we take ,1001 = 9

9mod 3 =0

And now check the transitions from initial state.

q0 on 1 goes to q1

q1 on 0 goes to q2

q2 on 0 goes to q1

q1 on 1 goes to q0.as it reached to final state .so this string is accepted .hope you got it ...