<Question>
Remmy wants to divide $10$ by $\frac{2}{3}$, but he cannot remember how to do that.  By what number should he multiply $10$ to get the answer?
</Question>

<think>
<Plan_1>
Okay, so Remmy needs to divide 10 by 2/3, but he's forget how to do that. Hmm, let me think. I remember that dividing by a fraction is the same as multiplying by its reciprocal. But wait, let me make sure I got that right. So, if you have a problem like 10 divided by 2/3, instead of doing 10 ÷ (2/3), you can flip the fraction 2/3 to become 3/2 and then multiply 10 by 3/2. That should give the correct answer.

But why does that work? Let me break it down.

<prompt_1.1> Explain why dividing by a fraction is equivalent to multiplying by its reciprocal. </prompt_1.1>

Let me test this with a simpler fraction to see if it makes sense. Suppose it's 1 divided by 1/2.

<prompt_1.2> Test the method of multiplying by the reciprocal with the example of 1 divided by 1/2. </prompt_1.2>

Another example: 5 divided by 1/4.
Using the same method, 5 * 4 = 20. And 5 ÷ (1/4) is indeed 20 because 1/4 of 20 is 5.

So going back to the original problem, 10 divided by 2/3.

<prompt_1.3> Apply the method to the original problem of dividing 10 by 2/3 and determine the multiplier. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
When you divide by a fraction, you're essentially asking, "How many times does this fraction fit into the number"? So, 10 divided by 2/3 is asking how many 2/3s are there in 10. To find that, multiplying by the reciprocal (3/2) will give the correct number of times. 

Let me verify with an example. If I take 6 divided by 2/3, using the reciprocal method, it would be 6 * 3/2 = 9. Let's check manually. How many 2/3s are in 6? Well, each whole number has 3 thirds, so 6 wholes have 18 thirds. Each 2/3 is like two-thirds, so how many two-thirds are in 18 thirds? 18 divided by 2 is 9.
</execution_1.1>

<execution_1.2>
If I multiply 1 by 1/2, I get 1/2, but dividing 1 by 1/2 should actually be 2 because 1/2 goes into 1 two times. So, in that case, flipping the denominator 1/2 to 2 and multiplying by the numerator 1 gives 2, which is correct.
</execution_1.2>

<execution_1.3>
If we multiply 10 by 3/2, we get (10 * 3)/2 = 30/2 = 15. Let me check if that's correct. 2/3 of 15 should be 10. Calculating 2/3 * 15: (2 * 15)/3 = 30/3 = 10.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.3:
Wait, but the question is asking by what number should he multiply 10. So, the answer is 3/2.

But maybe the problem is phrased as "by what number should he multiply 10 to get the answer"? So, the multiplier is 3/2.
Alternatively, could there be another way to approach this? Let me think.

Another method: Let's set up the equation. Dividing 10 by 2/3 is equal to 10 * (3/2) = 15. So, the number he needs to multiply by is 3/2.

<prompt_2.1> Verify the solution using algebraic manipulation and reciprocal properties. </prompt_2.1>

Let me just confirm once more. If he multiplies 10 by 3/2, he gets 15.

<prompt_2.2> Confirm the solution by performing the multiplication and ensuring it aligns with the original division. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
So, the number is 3/2. 

Alternatively, if he divides 10 by 2/3, that's 10 ÷ (2/3) = (10 * 3)/2 = 30/2 = 15. Therefore, multiplying 10 by 3/2 gives 15, which is the result of the division. 

So, the key here is recognizing that dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of 2/3 is 3/2. Therefore, Remmy should multiply 10 by 3/2 to get the answer.
</execution_2.1>

<execution_2.2>
Then, since dividing by a fraction is multiplying by its reciprocal, that's exactly what he needs to do. So, yes, the multiplier is 3/2. 

Alternatively, if he were to divide 10 by 3/2, he would multiply by 2/3, but that's not the case here. The original problem is 10 divided by 2/3, so the reciprocal is 3/2. Therefore, multiplying 10 by 3/2 gives the correct result.
</execution_2.2>
</Execution_2>

<Final_answer>
Therefore, the number by which Remmy should multiply $10$ to get the answer is $\boxed{\frac{3}{2}}$.
</Final_answer>
</think>