<Question>
What is the remainder of 2004 divided by 12?
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the remainder when 2004 is divided by 12. Hmm, how do I approach this?

First, I remember that when you divide a number by 12, the remainder has to be less than 12. So the possible remainders are 0 through 11. My job is to figure out which one of those 2004 leaves.

Maybe I can start by doing the division directly. Let me try dividing 2004 by 12.

<prompt_1.1> Perform direct division of 2004 by 12 to find the remainder. </prompt_1.1>

Wait, but I want to make sure I didn't make a mistake there. Let me check another way.

Alternatively, I could break down 2004 into parts that are easier to divide by 12. Let's see, 2004 is the same as 2000 + 4. Maybe I can find the remainder of 2000 divided by 12 and then add the remainder of 4 divided by 12, which is just 4, and then take that total modulo 12.

<prompt_1.2> Break down 2004 into parts to verify the remainder when divided by 12. </prompt_1.2>

Another method: Maybe using modular arithmetic properties. I know that 2004 can be expressed as 12*167 + r, where r is the remainder. So if I solve for r, r = 2004 - (12*167). Let me compute 12*167.

<prompt_1.3> Use modular arithmetic to verify the remainder when 2004 is divided by 12. </prompt_1.3>

Alternatively, I could use the fact that 12 is 3*4, and use the Chinese Remainder Theorem. But that might be overcomplicating. Let me try it for fun.

First, find 2004 mod 3. The sum of the digits of 2004 is 2+0+0+4=6, which is divisible by 3, so 2004 mod 3 is 0. Then 2004 mod 4. Since 4 divides 100, the last two digits 04 form 4, which is divisible by 4. So 2004 mod 4 is 0. Then, since 2004 mod 3 = 0 and mod 4 = 0, then mod 12 (since 3 and 4 are coprime) would also be 0. Wait, that's another way to see it.
So 2004 is divisible by both 3 and 4, hence by 12. Therefore, remainder is 0.

But let me check with another approach. Maybe using negative numbers. For example, 2004 divided by 12. 12*160=1920. 2004-1920=84. Then 84 divided by 12 is 7, so total is 160+7=167, remainder 0. Yep, same result.

Alternatively, maybe using division algorithm: a = bq + r, where 0 ≤ r < b. Here, a=2004, b=12.

<prompt_1.4> Use the division algorithm to verify the remainder when 2004 is divided by 12. </prompt_1.4>
</Plan_1>

<Execution_1>
<execution_1.1>
12 times 160 is 1920, right? Because 12*100=1200, 12*60=720, so 1200+720=1920. Then 2004 minus 1920 is 84. So 2004 = 12*167 + r. Wait, no, 12*167 would be 12*(160+7) = 12*160 + 12*7 = 1920 + 84 = 2004. So then the remainder would be 0? That can't be right. Wait, 12*167 is 2004, so the remainder is 0?
</execution_1.1>

<execution_1.2>
Let's start with 2000 divided by 12. Hmm, 12*166 is 1992. Then 2000 - 1992 is 8. So the remainder of 2000 divided by 12 is 8. Then adding 4 gives 8 + 4 = 12. But 12 divided by 12 is 1 with remainder 0. So the total remainder would be 0.
</execution_1.2>

<execution_1.3>
12*160=1920, 12*7=84, so 12*167=1920+84=2004. Wait, that can't be right. If 12*167 is 2004, then the remainder would be 0.
</execution_1.3>

<execution_1.4>
So 2004 = 12*q + r. We need to find q and r. Let's compute q = floor(2004/12). As before, 12*167=2004, so q=167, r=0.
</execution_1.4>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2, execution_1.3, execution_1.4:
But wait, that seems straightforward, but maybe I made a mistake here. Let me verify with another example. Let's take a number like 24.

<prompt_2.1> Verify the result by checking a simpler number, such as 24 divided by 12. </prompt_2.1>

Another check: 12*160=1920, 2004-1920=84. Then 84/12=7, so 160+7=167, remainder 0.

Alternatively, using a calculator: 2004 ÷ 12 = 167 exactly, so remainder 0.

Based on execution_1.2:
Wait, so all these methods point to the remainder being 0. Then why did I even doubt it initially? Maybe because 2004 is a multiple of 12? Let me check.

<prompt_2.2> Recheck if 2004 is a multiple of 12, confirming the remainder is 0. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Divided by 12, remainder 0. Correct. 13*12=156, remainder 0. Wait, 156 divided by 12 is 13, remainder 0. So 156 is 13*12. Similarly, 2004 divided by 12 is 167*12, so remainder 0.
</execution_2.1>

<execution_2.2>
Let's see, 12*170=2040, which is higher than 2004. So 12*160=1920, subtract that from 2004: 2004-1920=84. Then 84/12=7, so 12*7=84. So total is 160+7=167, 12*167=2004. Yes, exactly. Therefore, 2004 is divisible by 12, so the remainder is 0.
</execution_2.2>
</Execution_2>

<Final_answer>
Therefore, the remainder of 2004 divided by 12 is $\boxed{0}$.
</Final_answer>
</think>