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

We have three positive integers $a$, $b$, and $c$ with these relationships:

$ab = 432$

$bc = 96$

$c < 9$

We need to find the smallest possible value of $a + b + c$.

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Since we have two equations connecting three variables, let's use one equation to express one variable in terms of another, then substitute.

From $bc = 96$, we can write: $b = \tfrac{96}{c}$

Key Insight: For $b$ to be a positive integer, $c$ must be a divisor of 96.

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Let's find the divisors of 96:

$96 = 2^5 \times 3 = 32 \times 3$

The divisors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

Since $c < 9$, our possible values are: $c \in \{1, 2, 3, 4, 6, 8\}$

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Let's check each possible value of $c$:

$c = 1$, $b = 96$

$a = \tfrac{432}{96} = 4.5$ (not an integer)

$c = 2$, $b = 48$

$a = \tfrac{432}{48} = 9$

$c = 3$, $b = 32$

$a = \tfrac{432}{32} = 13.5$ (not an integer)

$c = 4$, $b = 24$

$a = \tfrac{432}{24} = 18$

$c = 5$, $b = 16$

$a = \tfrac{432}{16} = 27$

$c = 8$, $b = 12$

$a = \tfrac{432}{12} = 36$

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Cases where $c = 1, 3$ don't work because they make $a$ non-integer.

Valid combinations:

c	a + b + c
2	59
4	46
6	49
8	56

Therefore, the smallest possible value of $a + b + c$ is $46$.