Suppose a,b,c are three distinct natural numbers, such that 3ac = 8(a + b) . Then, the smallest possible value of 3a + 2b + c is

The correct answer is 12.

We are given that $a$, $b$, and $c$ are three distinct natural numbers satisfying the equation:

$3ac=8(a+b)$

We want to find the smallest possible value of the expression $3a+2b+c$.

First, let us express $b$ in terms of $a$ and $c$:

$3ac=8a+8b$

$8b=3ac-8a=a(3c-8)$

$b=\frac{a(3c-8)}{8}$

Since $b$ is a positive integer (natural number), we must have $3c-8>0$, which implies $3c>8$, so $c\ge 3$ (i.e. $c\ge 3$ in standard unicode: $c\ge 3$).

Now, substitute $2b=\frac{a(3c-8)}{4}$ into the target expression $3a+2b+c$:

$3a+2b+c=3a+\frac{a(3c-8)}{4}+c$

$=a\left(3+\frac{3c-8}{4}\right)+c$

$=\frac{a(3c+4)}{4}+c$

Now we test feasible integer values for $c$ and find valid distinct natural numbers $a$, $b$, and $c$:

1. If $c=3$:
$b=\frac{a(9-8)}{8}=\frac{a}{8}$.
Since $a$ must be a multiple of 8, the smallest natural number is $a=8$, which gives $b=1$.
Here, $(a,b,c)=(8,1,3)$ are distinct natural numbers.
The expression value is $3\left(8\right)+2\left(1\right)+3=24+2+3=29$.

2. If $c=4$:
$b=\frac{a(12-8)}{8}=\frac{a}{2}$.
To make $b$ a natural number, $a$ must be even.
If $a=2$, then $b=1$.
Here, $(a,b,c)=(2,1,4)$ are distinct natural numbers.
The expression value is $3\left(2\right)+2\left(1\right)+4=6+2+4=12$.

Testing other distinct combinations yields larger values (for instance, $c=8$, $a=1$, $b=2$ gives $3\left(1\right)+2\left(2\right)+8=15$).

Therefore, the smallest possible value of $3a+2b+c$ is 12.