Data Table

We have three positive integers $a$, $b$, and $c$ with these relationships: $$\newline$$ $ab = 432$ $$\newline$$ $bc = 96$ $$\newline$$ $c < 9$ $$\newline$$ We need to find the smallest possible value of $a + b + c$. $$\newline$$

$$\newline$$ Since we have two equations connecting three variables, let's use one equation to express one variable in terms of another, then substitute. $$\newline$$ From $bc = 96$, we can write: $b = \tfrac{96}{c}$ $$\newline$$ Key Insight: For $b$ to be a positive integer, $c$ must be a divisor of 96. $$\newline$$

$$\newline$$ Let's find the divisors of 96: $$\newline$$ $96 = 2^5 \times 3 = 32 \times 3$ $$\newline$$ The divisors of 96 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 $$\newline$$ Since $c < 9$, our possible values are: $c \in \{1, 2, 3, 4, 6, 8\}$ $$\newline$$

$$\newline$$ Let's check each possible value of $c$: $$\newline$$ $c = 1$, $b = 96$ $\newline$ $a = \tfrac{432}{96} = 4.5$ (not an integer) $$\newline$$ $c = 2$, $b = 48$ $\newline$ $a = \tfrac{432}{48} = 9$ $$\newline$$ $c = 3$, $b = 32$ $\newline$ $a = \tfrac{432}{32} = 13.5$ (not an integer) $$\newline$$ $c = 4$, $b = 24$ $\newline$ $a = \tfrac{432}{24} = 18$ $$\newline$$ $c = 5$, $b = 16$ $\newline$ $a = \tfrac{432}{16} = 27$ $$\newline$$ $c = 8$, $b = 12$ $\newline$ $a = \tfrac{432}{12} = 36$ $$\newline$$

$$\newline$$ Cases where $c = 1, 3$ don't work because they make $a$ non-integer. $$\newline$$ Valid combinations: $\newline$ $\newline$ $\newline$ Data Table $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$ $\newline$

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

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

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