Step 1: Express $b$ in terms of $a$ and $c$
Given: $3ac = 8a + 8b$
$8b = 3ac - 8a = a(3c - 8)$
$b = \frac{a(3c - 8)}{8}$
Step 2: Find possible values for $c$
Since $b \in \mathbb{N}$, then $3c - 8 > 0 \implies 3c > 8 \implies c \geq 3$.
Case 1: If $c = 3$, then $b = \frac{a(3(3)-8)}{8} = \frac{a}{8}$.
For $b$ to be a natural number, $a$ must be a multiple of $8$.
If $a=8$, then $b=1$. Numbers are $\{8, 1, 3\}$. All distinct natural numbers.
Value $= 3(8) + 2(1) + 3 = 24 + 2 + 3 = 29$.
Step 3: Test other combinations to minimize
Case 2: If $c = 4$, then $b = \frac{a(12-8)}{8} = \frac{4a}{8} = \frac{a}{2}$.
If $a=2$, then $b=1$. Numbers are $\{2, 1, 4\}$. Distinct.
Value $= 3(2) + 2(1) + 4 = 6 + 2 + 4 = 12$.
Case 3: If $c = 3$ and $a=16$, $b=2$. Value $= 3(16)+2(2)+3 = 55$ (Too large).
Case 4: If $c=6$, then $b = \frac{a(10)}{8} = \frac{5a}{4}$.
If $a=4$, then $b=5$. Numbers are $\{4, 5, 6\}$. Distinct.
Value $= 3(4) + 2(5) + 6 = 12 + 10 + 6 = 28$.
Case 5: If $c=5$, then $b = \frac{7a}{8}$. If $a=8, b=7$. $\{8, 7, 5\}$. Value $= 24+14+5 = 43$.
Step 4: Conclusion
The smallest value found among distinct natural numbers is 19 (wait, let's re-verify the $a=4, c=3$ set).
If $c=3, a=4, b=a(1)/8$ (Not integer).
Re-testing $c=4, a=6, b=3$. $\{6, 3, 4\}$. Distinct.
Sum $= 3(6) + 2(3) + 4 = 18 + 6 + 4 = 28$.
Checking $c=12, b = \frac{a(28)}{8} = \frac{7a}{2}$. If $a=2, b=7$. $\{2, 7, 12\}$. Sum $= 6+14+12=32$.
The minimum value occurs at $\{a=2, b=1, c=4\}$, giving $3(2)+2(1)+4 = \mathbf{12}$.
