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 the initial equation involving three distinct natural numbers a, b, and c:

$3ac=8(a+b)$

To understand the relationship between these variables, let's distribute the 8 on the right side and isolate b:

$3ac=8a+8b$

Subtracting 8a from both sides yields:

$8b=3ac-8a$

Now, we can factor out a on the right side:

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

Finally, dividing by 8 gives us an expression for b:

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

Because a, b, and c are natural numbers (positive integers), the value of b must be strictly positive. Since a is also positive, the expression inside the parentheses must be strictly greater than zero:

$3c-8>0$
$3c>8$

Since c must be an integer, the smallest possible value for c is 3.

Our goal is to minimize the value of the expression 3a + 2b + c. To find the smallest sum, we should systematically test the smallest possible values for c.

Case 1: Let c = 3

Substitute c = 3 into our factored equation for 8b:

$8b=a\left(3\right(3)-8)$
$8b=a(9-8)$
$8b=a$

To keep the numbers as small as possible, we choose the smallest possible natural number for b, which is 1. If b = 1, then a = 8. This gives us the numbers a = 8, b = 1, and c = 3, which are all distinct natural numbers. Substituting these into our expression yields:

$3a+2b+c=3\left(8\right)+2\left(1\right)+3=24+2+3=29$

Case 2: Let c = 4

Now let's substitute the next smallest value, c = 4, into the equation:

$8b=a\left(3\right(4)-8)$
$8b=a(12-8)$
$8b=4a$

Dividing both sides by 4 simplifies this to:

$2b=a$

Once again, picking the smallest natural number for b = 1 gives us a = 2. These numbers are a = 2, b = 1, and c = 4, which are all distinct. Let's calculate the value of the expression for this set of numbers:

$3a+2b+c=3\left(2\right)+2\left(1\right)+4=6+2+4=12$

Since we are looking for the absolute minimum value using distinct positive integers, 12 is a very small number and significantly lower than our previous result of 29. Let's quickly verify one more case to ensure the sum doesn't decrease further. If we let c = 8, the equation becomes 8b = 16a, meaning b = 2a. Using a = 1 yields b = 2 and c = 8. The sum would be 3(1) + 2(2) + 8 = 15, which is already larger than 12.

Thus, the minimum sum is achieved precisely when a = 2, b = 1, and c = 4, making the smallest possible value exactly 12.