CAT 2025 Slot 2 QA Q9 Number Systems

Question 9 of the Quantitative Ability (QA) section, CAT 2025 Slot 2, with CAT's official answer and a worked solution.

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

TITA: type in the answer. There are no options and no negative marking.

Answer: 12 Number Systems TITA

Solution

Rewrite: \(b = \frac{3ac}{8} - a = \frac{a(3c - 8)}{8}\), so \(c \ge 3\).
\(a = 1\): need \(8 \mid 3c - 8\), so \(c = 8\), \(b = 2\): value \(3 + 4 + 8 = 15\).
\(a = 2\): \(b = \frac{3c-8}{4}\), need \(4 \mid c\): \(c = 4\), \(b = 1\) (all distinct): value \(6 + 2 + 4 = 12\).
\(a = 3\) forces \(c\) to be a multiple of 8 (value already above 12), and \(a \ge 4\) makes \(3a + 2b + c > 12\).
Check: \(3 \cdot 2 \cdot 4 = 24 = 8(2+1)\). Minimum is \(12\).