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How many integer triplets (alpha, beta, gamma) satisfy all of the following simultaneously: 2*alpha - beta + 3*gamma = 4, alpha + beta - 3*gamma = -1, 5*alpha - beta + 3*gamma = 7, alpha*beta*gamma <= 0, with alpha, beta, gamma integers?

1. 0
2. 1
3. 2
4. 3

Correct answer: 0

Solution

Equations 1 and 3 give 3*alpha = 3, so alpha = 1, but substituting back into equations 1 and 2 leads to an inconsistency, so the system has no solution and no triplet exists.

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