The number of all five digit numbers which are divisible by 4 that can be formed from the digits 0,1,2,3,4 (with repetition) is

Options

A. 36
B. 30
C. 34
D. none of these

Answer

To find the number of all five-digit numbers that can be formed from the digits 0, 1, 2, 3, and 4 (with repetition) and are divisible by 4, we can follow these steps:

### Step 1: Understand the divisibility rule for 4
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

### Step 2: Identify valid last two-digit combinations
From the digits 0, 1, 2, 3, and 4, we need to find the combinations of the last two digits that are divisible by 4. The valid combinations are:
- 00
- 12
- 20
- 24
- 32
- 40

### Step 3: Count the valid last two-digit combinations
We have identified the following valid combinations for the last two digits:
1. 00
2. 12
3. 20
4. 24
5. 32
6. 40

This gives us a total of 6 valid combinations.

### Step 4: Determine the choices for the first three digits
The first digit of a five-digit number cannot be 0 (to ensure it remains a five-digit number). Therefore, the choices for the first digit can be 1, 2, 3, or 4 (4 options).

For the second and third digits, we can use any of the digits (0, 1, 2, 3, 4) since repetition is allowed. Therefore, there are 5 options for each of these digits.

### Step 5: Calculate the total combinations
- For the first digit: 4 choices (1, 2, 3, 4)
- For the second digit: 5 choices (0, 1, 2, 3, 4)
- For the third digit: 5 choices (0, 1, 2, 3, 4)

Thus, the total number of combinations for the first three digits is:
\[ 4 \times 5 \times 5 = 100 \]

### Step 6: Combine the choices
Now, we combine the choices for the last two digits (6 valid combinations) with the choices for the first three digits (100 combinations):
\[ \text{Total five-digit numbers} = 100 \times 6 = 600 \]

### Final Answer
The total number of all five-digit numbers that can be formed from the digits 0, 1, 2, 3, and 4 (with repetition) and are divisible by 4 is **600**.

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