Pove that the semigroup s 4n 5n 6n is symmetric if s

Question

Question: Prove that the semigroup S = 4N + 5N + 6N is symmetric ? If S = 5N + 6N + 7N + 8N and K is a field, prove that K[S] is a Gorenstein ring which is not a complete intersection ? Let S ⊂ N be a numerical semigroup with Frobenius number c and let K be a field. If D = K[S] ⊂ K[x] has the induced grading, prove that the degree of the Hilbert series of D/x^c+1D is 2c + 1.