$(n -1)(n - 3)(n - 5) \cdots (n -2013) = n(n + 2)(n + 4) \cdots(n + 2012)$

How many integer solutions are there for the equation above?

The answer is 0.

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If n is odd, the LHS is even or zero and the RHS is odd.

If n is even, the LHS is odd and the RHS is even or zero.

So there is no integer solution for the equation.