A number rounded up to the next thousand is 8000. It is divisible by 6 and 10 but not by 9. The hundreds digit is three more than the tens digit. What is the number?

Rather than use algebra, I would use trial and error. The number is between 7500 and 8499 and ends with zero. Possibilities that are also divisible by 6 are:

7500, 7530, 7560, 7590, 7620, 7630, 7680, 7710, 7740, 7770, 7800, 7830, 7860, 7890, 7920, 7950, 7980, 8010, 8040, 8070, 8100, 8130, 8160, 8190, 8220, 8250, 8280, 8310, 8340, 8370, 8400, 8430, and 8460.

The only ones that meet the last requirement -- about the hundreds and tens digits -- are 7630 and 7740. Only 7740 is not evenly divisible by 9; so that is the answer.

To solve this problem, we need to work through the given information step by step.

1. A number rounded up to the next thousand is 8000.
- This means that the number is between 7000 and 8000. Let's keep this in mind.

2. The number is divisible by 6 and 10.
- Since the number is divisible by 6, it must also be divisible by 2 and 3.
- Since the number is divisible by 10, it must end with a 0.

3. The number is not divisible by 9.
- This means that the sum of its digits is not divisible by 9.
- Let's calculate the sum of the digits: 8 + 0 + 0 + 0 = 8.
- Since 8 is not divisible by 9, the number is not divisible by 9 either.

4. The hundreds digit is three more than the tens digit.
- This means that if we represent the number as "XYZ0" (where X is thousands, Y is hundreds, Z is tens), then Y = Z + 3.

Now, let's start with the thousands digit. Since the number is rounded up to the next thousand and it is between 7000 and 8000, the thousands digit must be 7.

Next, let's focus on the hundreds and tens digits. Since Y = Z + 3, there are a few possibilities:

If Z = 0, then Y = 3. But this leads to the number being 7300, which does not satisfy the condition of being rounded up to the next thousand.

If Z = 1, then Y = 4. But this leads to the number being 7410, which is not divisible by 6.

If Z = 2, then Y = 5. But this leads to the number being 7520, which is also not divisible by 6.

Finally, if Z = 3, then Y = 6. This results in the number being 7630, which is divisible by 6, 10, and not by 9.

Therefore, the number that satisfies all the given conditions is 7630.