Given the equations \( ab = 432 \), \( bc = 96 \), and \( c < 9 \). Determine the minimum value of \( a + b + c \).
From the equations, we express \( a \) in terms of \( b \) and \( c \): \( a = \frac{432}{b} \) and \( b = \frac{96}{c} \).
Substituting the expression for \( b \) into the equation for \( a \):
\( a = \frac{432}{\frac{96}{c}} = \frac{432c}{96} = \frac{9c}{2} \)
For \( a \) to be an integer, \( \frac{9c}{2} \) must be an integer. This requires \( c \) to be an even number.
Given the constraint \( c < 9 \), the possible even integer values for \( c \) are 2, 4, 6, and 8.
Calculate \( a + b + c \) for each valid value of \( c \):
| c | a | b | a+b+c |
|---|---|---|---|
| 2 | \( \frac{9 \times 2}{2} = 9 \) | \( \frac{96}{2} = 48 \) | 9 + 48 + 2 = 59 |
| 4 | \( \frac{9 \times 4}{2} = 18 \) | \( \frac{96}{4} = 24 \) | 18 + 24 + 4 = 46 |
| 6 | \( \frac{9 \times 6}{2} = 27 \) | \( \frac{96}{6} = 16 \) | 27 + 16 + 6 = 49 |
| 8 | \( \frac{9 \times 8}{2} = 36 \) | \( \frac{96}{8} = 12 \) | 36 + 12 + 8 = 56 |
The smallest value for \( a + b + c \) is 46, occurring when \( c = 4 \).