Introduction:
To find the smallest 4-digit number that is divisible by 18, 24, and 32, we need to determine the common multiples of these three numbers. We can start by finding the least common multiple (LCM) of 18, 24, and 32 and then look for the smallest 4-digit number greater than or equal to this LCM.

Step 1: Finding the LCM:
To find the LCM of 18, 24, and 32, we can use the prime factorization method. We express each number as a product of prime factors:

18 = 2 * 3^2
24 = 2^3 * 3
32 = 2^5

We take the highest power of each prime factor that appears in any of the numbers:

2^5 * 3^2 = 288

Therefore, the LCM of 18, 24, and 32 is 288.

Step 2: Finding the smallest 4-digit number:
To find the smallest 4-digit number that is greater than or equal to 288, we need to determine the smallest 4-digit number. The smallest 4-digit number is 1000.

Step 3: Checking divisibility:
Now, we need to check if 1000 is divisible by 18, 24, and 32.

To check if a number is divisible by 18, we need to check if it is divisible by both 2 and 9. 1000 is divisible by 2 because it ends with a 0. To check if it is divisible by 9, we find the sum of its digits: 1 + 0 + 0 + 0 = 1. Since the sum of its digits is not divisible by 9, 1000 is not divisible by 18.

To check if a number is divisible by 24, we need to check if it is divisible by both 2 and 3. 1000 is divisible by 2 because it ends with a 0. To check if it is divisible by 3, we find the sum of its digits: 1 + 0 + 0 + 0 = 1. Since the sum of its digits is not divisible by 3, 1000 is not divisible by 24.

To check if a number is divisible by 32, we need to check if it is divisible by 2^5. 1000 is not divisible by 2^5 because it does not have 5 consecutive zeros at the end.

Conclusion:
The smallest 4-digit number that is divisible by 18, 24, and 32 is 1000.