The number of all four digit numbers which are divisible by 4 that can be formed from the digits 1, 2, 3, 4, and 5, is

Options

A. 125
B. 30
C. 95
D. none of these

Answer

To solve the problem of finding the number of four-digit numbers that can be formed from the digits 1, 2, 3, 4, and 5, which 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. Therefore, we need to focus on the last two digits of our four-digit number.

### Step 2: Identify valid pairs for the last two digits
We need to find all possible pairs of the last two digits from the digits 1, 2, 3, 4, and 5 that are divisible by 4. The valid pairs are:
- 12 (12 ÷ 4 = 3)
- 24 (24 ÷ 4 = 6)
- 32 (32 ÷ 4 = 8)
- 44 (44 ÷ 4 = 11)
- 52 (52 ÷ 4 = 13)

So, the valid pairs of last two digits are: 12, 24, 32, 44, and 52.

### Step 3: Count the number of choices for the last two digits
We have identified 5 valid pairs for the last two digits. Therefore, there are 5 choices for the last two digits.

### Step 4: Count the choices for the first two digits
Since the digits can be repeated, we can choose any of the 5 digits (1, 2, 3, 4, 5) for the first two digits. Thus, for each of the first two digits, we have:
- 5 choices for the first digit
- 5 choices for the second digit

### Step 5: Calculate the total number of four-digit numbers
Now, we can calculate the total number of four-digit numbers that can be formed:
- Total choices for the last two digits = 5
- Total choices for the first digit = 5
- Total choices for the second digit = 5

The total number of four-digit numbers is given by:
\[ \text{Total} = (\text{Choices for first digit}) \times (\text{Choices for second digit}) \times (\text{Choices for last two digits}) \]
\[ \text{Total} = 5 \times 5 \times 5 = 125 \]

### Final Answer
The total number of four-digit numbers that can be formed from the digits 1, 2, 3, 4, and 5, which are divisible by 4, is **125**.

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