How many positive even three-digit

integers are divisible by 3?

Alright,

The answer is 150.
Like if it is correct!

To find the number of positive even three-digit integers that are divisible by 3, we can break down the problem into two parts:

1. Finding the total number of three-digit integers:
The three-digit integers range from 100 to 999. To find the count, subtract the lowest possible three-digit integer from the highest possible three-digit integer and add 1.
999 - 100 + 1 = 900

2. Finding the number of three-digit integers divisible by 3:
A number is divisible by 3 if the sum of its digits is divisible by 3. Since we only want to consider even numbers, the last digit must be divisible by 2, which means it can only be 0, 2, 4, 6, or 8.

- Case 1: The last digit is 0:
The other two digits can be any combination from 1 to 9 (excluding 0), which is 9 choices.
So, the count for this case is 9.

- Case 2: The last digit is 2:
The other two digits can be any combination from 0 to 8 (excluding 2), which is 8 choices.
So, the count for this case is 8.

- Case 3: The last digit is 4:
The other two digits can be any combination from 0 to 8 (excluding 4), which is 8 choices.
So, the count for this case is 8.

- Case 4: The last digit is 6:
The other two digits can be any combination from 0 to 8 (excluding 6), which is 8 choices.
So, the count for this case is 8.

- Case 5: The last digit is 8:
The other two digits can be any combination from 0 to 8 (excluding 8), which is 8 choices.
So, the count for this case is 8.

Therefore, the total count for the even three-digit numbers divisible by 3 is the sum of all the case counts:
9 + 8 + 8 + 8 + 8 = 41

So, there are 41 positive even three-digit integers that are divisible by 3.