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Lets first prove the number of states in the minimal DFA for accepting binary strings divisible by a given number say $12$ We need $3$ states for checking if a binary number is divisible by $3$ - each state corresponding to remainders $0,1,2.$ Here, remainder $0$ will be the final state for divisibility by $3$.

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However, a number divisible by 3 is not necessarily divisible by 9. For example 6, 12, 15, 21, 24, 30 are all divisible by 3 but none of them is divisible by 9. The rule for divisibility by 3 can be easily obtained following the same logic we used with divisibility by 9.

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And so the domain of this function is really all positive integers - N has to be a positive integer. And so we can try this out with a few things, we can take S of 3, this is going to be equal to 1 plus 2 plus 3, which is equal to 6. We could take S of 4, which is going to be 1 plus 2 plus 3 plus 4, which is going to be equal to 10.

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a) Construct the Finite Automata for binary umber divisible by 2 b) Design FA for decimal number divisible by 5 c) Give formal definition of Turing Machine d) State and explain closure properties of regular languages e) Construct DFA accepting all the strings corresponding to the Regular expression Q2. a) Construct the following grammar to CNF

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Construct a DFA to accept all strings which satisfies #(x) mod 5=2 Construct a DFA to accept all strings (0+1)* with an equal number of 0's & 1's such that each prefix has at most one more zero than ones and at most one more one than zeroes

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2. Show that the set of binary integers (given as strings over f0;1g) that are divisible by 3 is regular, by giving a DFA that recognizes it. Leading 0s are allowed. The empty string should be accepted. Brieﬂy explain your answer. 3. Let = fa;bg, let kbe a positive integer constant, and let L k be the language deﬁned as follows. L

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Divisible by n , so there should be n states starting from 0 to (n-1). let's take m % n = 0 , where m is the input number . so , the final state should be 0 .( as Similarly in the same manner DFA for binary number divisible by 3 will be : Now , let's see another type of question where remainder is not zero...