<s>[INST] 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 Below are the steps I have done so far.
Find $a$ in terms of $c$
$$
ab = 432 \\
bc = 96 \\
\frac{ab}{bc} = \frac ac = \frac 92 \implies a = 4.5c.
$$
Find $b$ in terms of $c$
$$  
ab = 432 \\
\implies 4.5c \cdot b = 432 \\
\implies b = \frac{96}c.
$$
Substituting, we get
$$
a + b + c = 4.5c + \frac{96}c + c = 5.5c + \frac{96}c.
$$
How should I proceed from here?
 [/INST] $bc=96$ and $c<9$ constrains $c$ to be one of $1,2,3,4,6,8$. Since $\gcd(432,96)=48$, $c$ cannot be $1$ or $3$, leaving four cases: $(a,b,c)=(9,48,2),(18,24,4),(27,16,6),(36,12,8)$. Of these, $(18,24,4)$ achieves the smallest $a+b+c$ of $46$.
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