CAT 2025 Slot 2 QA Question 12: Suppose $a, b, c$ are three distinct natural numbers, such that $3ac = 8(a+b)$. Then, the smallest possible value of $3a + 2b + c$ is

Question

Suppose $a, b, c$ are three distinct natural numbers, such that $3ac = 8(a+b)$. Then, the smallest possible value of $3a + 2b + c$ is

Accepted Answer

$3ac = 8(a+b)$
$\Rightarrow b = \dfrac{a(3c-8)}{8}$

Since $b$ is a positive natural number, $c$ must be at least $3$.

If $a \geq 3$, then $3a + 2b + c$ is at least $12$, so any value below $12$ can only occur for $a = 1$ or $a = 2$.

For $a = 1$:
$3c = 8(1+b)$
The smallest feasible positive soluti

Suppose a,b,ca, b, ca,b,c are three distinct natural numbers, such that 3ac=8(a+b)3ac = 8(a+b)3ac=8(a+b). Then, the smallest possible value of 3a+2b+c3a + 2b + c3a+2b+c is