Find the number of even positive integers which have three digits?

Answer

To find the number of even positive integers that have three digits, we can break down the problem into a series of steps:

### Step 1: Identify the range of three-digit integers
Three-digit integers range from 100 to 999.

### Step 2: Determine the last digit
Since we are looking for even integers, the last digit must be one of the even digits. The even digits available are: 0, 2, 4, 6, and 8. However, since we are dealing with three-digit numbers, the last digit cannot be 0 (as it would not contribute to being a three-digit number). Therefore, the possible last digits are: 2, 4, 6, and 8. This gives us **4 options** for the last digit.

### Step 3: Determine the first digit
The first digit of a three-digit number cannot be 0 (otherwise it would be a two-digit number). The possible digits for the first position are: 1, 2, 3, 4, 5, 6, 7, 8, and 9. This gives us **9 options** for the first digit.

### Step 4: Determine the middle digit
The middle digit can be any digit from 0 to 9. Therefore, there are **10 options** for the middle digit.

### Step 5: Calculate the total number of even three-digit integers
To find the total number of even three-digit integers, we multiply the number of choices for each digit together:
- Number of choices for the first digit = 9
- Number of choices for the middle digit = 10
- Number of choices for the last digit = 4

Thus, the total number of even three-digit integers is:
\[ \text{Total} = (\text{choices for first digit}) \times (\text{choices for middle digit}) \times (\text{choices for last digit}) \]
\[ \text{Total} = 9 \times 10 \times 4 = 360 \]

### Final Answer
The number of even positive integers which have three digits is **360**.

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