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We can arrive at the solution fairly easily by analysis. But this Python program checks all possible values for the letters. It runs in 3.8s (under PyPy).

Solution:Five of the equations must be discarded. Only the last one has solutions.Analytically we can see that:

simplifies to:

and any assignment of the remaining digits to E, N, I, G, M, A will give a valid equation. So there are 9!/(9-6)! = 60480 possibilities (before additional constraints, such as disallowing leading zeros, are added).

Similarly the remaining equations can be rewritten as:

The first and second equations are obviously not going to have solutions.

For the third equation GMA/ENI would be a 1 or 2 digit number, making GMAS = 10xy, but then GMA = 10x and there is no possible assignment of ENI that would make this work.

A similar argument eliminates the fourth and fifth equations from having solutions.

Hence only the last equation given in the puzzle has solutions, and they are all trivial.