**Solution:**

To find the remainder when N is divided by 9, we need to consider the divisibility rule for 9.

According to the divisibility rule for 9, a number is divisible by 9 if the sum of its digits is divisible by 9.

Let's consider the number N = 1000x + 100y + 10z.

**Step 1: Find the sum of the digits of N**

The sum of the digits of N can be calculated as:

sum = x + y + z

**Step 2: Find the remainder when sum is divided by 9**

To find the remainder when sum is divided by 9, we can use the fact that the remainder when a number is divided by 9 is the same as the remainder when the sum of its digits is divided by 9.

Therefore, we need to find the remainder when sum is divided by 9.

**Step 3: Find the possible values of x, y, and z**

From the given information, we know that x, y, and z are different positive integers less than 4.

The possible values of x, y, and z are:

x = 1, 2, 3
y = 1, 2, 3
z = 1, 2, 3

**Step 4: Find the possible values of sum**

Using the values of x, y, and z, we can find the possible values of sum.

For x = 1, y = 1, z = 1, sum = 1 + 1 + 1 = 3
For x = 1, y = 1, z = 2, sum = 1 + 1 + 2 = 4
For x = 1, y = 1, z = 3, sum = 1 + 1 + 3 = 5
For x = 1, y = 2, z = 1, sum = 1 + 2 + 1 = 4
For x = 1, y = 2, z = 2, sum = 1 + 2 + 2 = 5
For x = 1, y = 2, z = 3, sum = 1 + 2 + 3 = 6
For x = 1, y = 3, z = 1, sum = 1 + 3 + 1 = 5
For x = 1, y = 3, z = 2, sum = 1 + 3 + 2 = 6
For x = 1, y = 3, z = 3, sum = 1 + 3 + 3 = 7
For x = 2, y = 1, z = 1, sum = 2 + 1 + 1 = 4
For x = 2, y = 1, z = 2, sum = 2 + 1 + 2 = 5
For x = 2, y = 1, z = 3, sum = 2 + 1 + 3 = 6
For x = 2, y = 2, z = 1, sum = 2 + 2 + 1 = 5
For x =