CAT 2020 Slot 1 QA Question 2: If $a$, $b$ and $c$ are positive integers such that $ab = 432$, $bc = 96$ and $c

Question

If $a$, $b$ and $c$ are positive integers such that $ab = 432$, $bc = 96$ and $c < 9$, then the smallest possible value of $a + b + c$ is

Accepted Answer

We can relate a and c by dividing the two given products.

$\dfrac{ab}{bc} = \dfrac{432}{96}$
$\Rightarrow \dfrac{a}{c} = \dfrac{9}{2}$
So, let $a = 9k$ and $c = 2k$.

From $bc = 96$, we get $b = \dfrac{96}{2k} = \dfrac{48}{k}$.

Since $c

$a + b + c = 9k + \dfrac{48}{k} + 2k = 11k + \dfrac{48}{k}$

If aaa, bbb and ccc are positive integers such that ab=432ab = 432ab=432, bc=96bc = 96bc=96 and c<9c < 9c<9, then the smallest possible value of a+b+ca + b + ca+b+c is