The number of 5 digit numbers which are divisible by 4, with digits from the set `{1,\ 2,\ 3,\ 4,\ 5}` and the repetition of digits is allowed, is ________.

Answer

To solve the problem of finding the number of 5-digit numbers that are divisible by 4, using digits from the set {1, 2, 3, 4, 5} with repetition allowed, 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 5-digit number.

### Step 2: Identify valid pairs of last two digits
We will find all the two-digit combinations that can be formed using the digits from the set {1, 2, 3, 4, 5} and check which of these are divisible by 4.

- Possible two-digit combinations:
- 11, 12, 13, 14, 15
- 21, 22, 23, 24, 25
- 31, 32, 33, 34, 35
- 41, 42, 43, 44, 45
- 51, 52, 53, 54, 55

Now we check which of these combinations are divisible by 4:
- 12 (12 ÷ 4 = 3)
- 24 (24 ÷ 4 = 6)
- 32 (32 ÷ 4 = 8)
- 44 (44 ÷ 4 = 11)
- 52 (52 ÷ 4 = 13)

Thus, the valid pairs of last two digits that are divisible by 4 are: **12, 24, 32, 44, 52**.

### Step 3: Count the valid last two digits
We found 5 valid pairs of last two digits that make the number divisible by 4.

### Step 4: Determine the choices for the first three digits
Since repetition of digits is allowed, each of the first three digits can be any of the 5 digits from the set {1, 2, 3, 4, 5}.

- Choices for the first digit: 5
- Choices for the second digit: 5
- Choices for the third digit: 5

### Step 5: Calculate the total number of combinations
The total number of 5-digit numbers can be calculated by multiplying the number of choices for the first three digits by the number of valid pairs of last two digits.

Total combinations = (Choices for the first digit) × (Choices for the second digit) × (Choices for the third digit) × (Valid pairs of last two digits)

Total combinations = 5 × 5 × 5 × 5 = 625

### Final Answer
The total number of 5-digit numbers that can be formed under the given conditions is **625**.

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