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Let Delta(alpha, beta, gamma) be the determinant: |3*alpha², alpha²+alpha*beta+beta², alpha²+alpha*gamma+gamma²; alpha²+alpha*beta+beta², 3*beta², beta²+beta*gamma+gamma²; alpha²+alpha*gamma+gamma², beta²+beta*gamma+gamma², 3*gamma²|. If alpha, beta, gamma are distinct primes, find the minimum value of |Delta(alpha, beta, gamma)|.
- 25
- 49
- 16
- 36
Correct answer: 36
Solution
The determinant factors as Delta = (alpha-beta)²*(beta-gamma)²*(gamma-alpha)². Minimum over distinct primes (2,3,5): 1*4*9 = 36.
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