Solution:

Given, ax - 3a = -12

We need to find the integer value of a for which x is a positive integer.

Let's solve the given equation step by step:

Step 1: Factorising

Taking common factor a from the left-hand side of the equation, we get:

a(x - 3) = -12

Step 2: Solving for x

Dividing both sides of the equation by a, we get:

x - 3 = -12/a

Adding 3 to both sides of the equation, we get:

x = -12/a + 3

Step 3: Finding the value of a for which x is a positive integer

For x to be a positive integer, the value of -12/a + 3 must be a positive integer.

So, we can write:

-12/a + 3 = n, where n is a positive integer

Solving for a, we get:

-12/a = n - 3

a = -12/(n - 3)

To find the integer value of a, we need to find the values of n for which a is an integer.

To do this, we can check the factors of -12 and see which ones give integer values of a.

The factors of -12 are: -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, 12.

For n = 1, a = -12/(1 - 3) = 6, which is an integer.

For n = 2, a = -12/(2 - 3) = -12, which is an integer.

For n = 3, a is undefined as we cannot divide by 0.

For n = 4, a = -12/(4 - 3) = -12, which is an integer.

For n = 5, a = -12/(5 - 3) = 6, which is an integer.

For n = 6, a is undefined as we cannot divide by 0.

For n = 7, a = -12/(7 - 3) = 3, which is an integer.

For n = 8, a = -12/(8 - 3) = -2.4, which is not an integer.

For n = 9, a = -12/(9 - 3) = -1.5, which is not an integer.

For n = 10, a = -12/(10 - 3) = -1.714, which is not an integer.

For n = 11, a = -12/(11 - 3) = -1.2, which is not an integer.

For n = 12, a = -12/(12 - 3) = -1.09, which is not an integer.

Therefore, the integer values of a for which x is a positive integer are 6, -12, and 6.