GATE | GATE-CS-2006 | Question 85

For S ∈ (0 + 1) * let d(s) denote the decimal value of s (e.g. d(101) = 5). Let L = {s ∈ (0 + 1)* d(s)mod5 = 2 and d(s)mod7 != 4}.
Which one of the following statements is true?

(A) L is recursively enumerable, but not recursive
(B) L is recursive, but not context-free
(C) L is context-free, but not regular
(D) L is regular

Answer: (D)

Explanation: It is regular
L1=d(s) mod 5 =2 is regular with 5 states
L2=d(s) mod 7 =4 is regular with 7 states
therefore L1 ^ L2′ should be regular
because regular grammar are closed under intersection and compliment

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