Case 1: Last 2 digits include 0 
Number options for the last 2 digits = 3. (04, 20 or 40.) 
Number of ways to arrange the remaining 4 digits = 4! = 24. 
To combine these options, we multiply: 
3*24 = 72. 

Case 2: Last 2 digits do NOT include 0 
Number options for the last 2 digits = 4. (12, 24, 32, or 52.) 
Number of options for the ten-thousands place = 3. (Any of the remaining 4 digits but 0.) 
Number of ways to arrange the remaining 3 digits = 3! = 6. 
To combine these options, we multiply: 
4*3*6 = 72. 

Total options = 72+72 = 144.

Understanding Divisibility by 4
To determine the number of 5-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4, and are divisible by 4, we must focus on the last two digits of the number. A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
Possible Last Two Digits
We will evaluate the combinations of the last two digits using the available digits:
- 00 (divisible)
- 12 (divisible)
- 20 (divisible)
- 24 (divisible)
- 32 (divisible)
- 40 (divisible)
Thus, the valid pairs of last two digits that are divisible by 4 are: 00, 12, 20, 24, 32, and 40.
Counting the Combinations
1. Last Two Digits:
- There are 6 valid combinations of the last two digits.
2. First Three Digits:
- The first digit cannot be 0 (to ensure it's a 5-digit number). Therefore, it can be 1, 2, 3, or 4 (4 options).
- The second and third digits can be any of the digits 0, 1, 2, 3, or 4 (5 options each).
Total Combinations Calculation
- Choices for First Digit: 4 choices (1, 2, 3, 4)
- Choices for Second Digit: 5 choices (0, 1, 2, 3, 4)
- Choices for Third Digit: 5 choices (0, 1, 2, 3, 4)
Total Count of 5-Digit Numbers
The total number of valid 5-digit numbers can be calculated as follows:
- Total = (Choices for First Digit) × (Choices for Second Digit) × (Choices for Third Digit) × (Choices for Last Two Digits)
Therefore:
Total = 4 × 5 × 5 × 6 = 600
Final Answer
The total number of 5-digit numbers using the digits 0, 1, 2, 3, and 4 that are divisible by 4 is 600.