Q9
Suppose \( a , b , c \) are three distinct natural numbers, such that \( 3 a c = 8 ( a + b ) \). Then, the smallest possible value of \( 3 a + 2 b + c \) is
Answer: 12
Key idea: Express \(b\) in terms of \(a\) and \(c\), then try small values of \(a\).
From \(3ac = 8a + 8b\): \(b = \dfrac{a(3c-8)}{8}\). For \(b \ge 1\) we need \(3c > 8\), so \(c \ge 3\).
- \(a = 1\): \(8 \mid 3c - 8\) forces \(8 \mid c\); smallest \(c = 8\), \(b = 2\). Value \(= 3 + 4 + 8 = 15\).
- \(a = 2\): \(b = \dfrac{3c-8}{4}\) needs \(4 \mid c\); \(c = 4\) gives \(b = 1\). The numbers \(2, 1, 4\) are distinct. Value \(= 6 + 2 + 4 = 12\).
- \(a \ge 3\): \(3a \ge 9\), while \(2b + c \ge 2 + 3 = 5\), so the value is at least 14.
Check: \(3 \cdot 2 \cdot 4 = 24 = 8(2+1)\). The minimum is 12.
Hence, the answer is 12.