To solve the problem of finding the number of four-digit numbers formed using the digits 0, 2, 4, and 5 that are not divisible by 5, we can follow these steps:
### Step 1: Identify the digits and conditions
The available digits are 0, 2, 4, and 5. A four-digit number cannot start with 0. Additionally, since we want the number to be not divisible by 5, the last digit cannot be 0 or 5.
### Step 2: Determine the last digit
Since the number must not be divisible by 5, the last digit can only be 2 or 4. Thus, we have 2 choices for the last digit.
### Step 3: Determine the first digit
The first digit of a four-digit number cannot be 0. If the last digit is 2, the available digits for the first digit are 0 (not allowed), 4, and 5. If the last digit is 4, the available digits for the first digit are 0 (not allowed), 2, and 5. Therefore, in both cases, we have 2 choices for the first digit (either 2 or 5 if the last digit is 4, and either 4 or 5 if the last digit is 2).
### Step 4: Determine the second and third digits
After choosing the first digit and the last digit, we will have 2 digits already used. We can use any of the remaining digits (including 0) for the second and third positions. Since we have 4 digits total and 2 digits have already been used, we have 2 remaining digits to choose from for the second digit, and then 1 remaining digit for the third digit.
### Step 5: Calculate the total combinations
Now, we can calculate the total number of combinations:
- For each choice of the last digit (2 choices: 2 or 4),
- For each choice of the first digit (2 choices),
- For the second digit (2 choices),
- For the third digit (1 choice).
Thus, the total number of four-digit numbers is calculated as follows:
\[
\text{Total combinations} = (\text{choices for last digit}) \times (\text{choices for first digit}) \times (\text{choices for second digit}) \times (\text{choices for third digit})
\]
\[
= 2 \times 2 \times 2 \times 1 = 8
\]
### Final Answer
The total number of four-digit numbers formed by using the digits 0, 2, 4, and 5 that are not divisible by 5 is **8**.
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