\(+\) and \(\times\) defined

Algebra Level 4

Define \(S(n)\) to be the term directly succeeding \(n\) in A000027. For example, \(S(5) = 6\) because the term succeeding \(5\) in the sequence is \(6\). Also define \(S(0) = 1\). \(S\) is undefined everywhere else, and you may assume that \(S\) is injective.

Define the operation \(+\) as follows:

  • \(a + 0 = a\)
  • \(a + S(b) = S(a+b)\)

Define the operation \(\times\) as follows:

  • \(a \times 0 = 0\)
  • \(a \times S(b) = a + (a \times b)\)

Determine the value of \(116180912 \times 615151219\).

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