Let a digit be one of the ten decimal digits 0,1,2,3,4,5,6,7,8,9 . Draw the state diagram for a DFA that recognizes the language of digit strings that represent base-10 numbers that are one more than a multiple of three. For example, your DFA should accept 07658 since 07654 = 7654 = 3(2551) + 1. On the other hand, 7655 would not be accepted since 7655 = 3(2551) + 2.

Hint: Use three states, representing numbers of the form 3n, 3n+1 and 3n+2. Think about what happens when we add a digit to the end of a given number. For example, when we add a 0, 3n+1, for example, will go to 10(3n+1) = 3(10n+3)+1, in other words, back to a number of the form 3n+1. The same thing will happen when we add a 3, 6 or 9 to the end. Continuing in this way you can fill in all the transitions between the three states.

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