<Question>
What is the last nonzero digit to the right of the decimal point in the decimal expansion of $\frac{137}{500}$?
</Question>

<think>
<Plan_1>
Okay, let's see. The problem is asking for the last nonzero digit to the right of the decimal point in the decimal expansion of 137 divided by 500. Hmm. Alright, so first, maybe I should convert 137/500 into a decimal to see what it looks like.

<prompt_1.1> Convert the fraction 137/500 into a decimal and identify the last nonzero digit to the right of the decimal point. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Let me start by doing the division. 500 goes into 137 zero times, so we start with 0. and then divide 1370 by 500. Wait, maybe it's easier to convert the denominator to 1000 somehow? Because 500 is half of 1000. So maybe if I multiply both the numerator and denominator by 2, that would make the denominator 1000. Let me try that.

So, 137 times 2 is 274, and 500 times 2 is 1000. So 137/500 is equal to 274/1000. Then, 274 divided by 1000 is 0.274. So the decimal expansion is 0.274. Therefore, the digits to the right of the decimal point are 2, 7, and 4. The last nonzero digit would be 4.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, but that seems too straightforward.

Let me double-check.
Is 137 divided by 500 really 0.274? Let me do the division manually. 500 goes into 137 zero times. Add a decimal point and a zero, making it 1370. Now, 500 goes into 1370 two times because 500 times 2 is 1000. Subtract 1000 from 1370, we get 370. Bring down another zero, making it 3700. 500 goes into 3700 seven times (500*7=3500). Subtract 3500 from 3700, we get 200. Bring down another zero, making it 2000. 500 goes into 2000 four times (500*4=2000). Subtract that and we get zero.

So the decimal is indeed 0.274. So the digits after the decimal are 2, 7, 4. The last nonzero digit is 4. So maybe the answer is 4?

But the question says "the last nonzero digit to the right of the decimal point". In 0.274, the digits to the right of the decimal are 2, 7, 4. So the last one is 4, which is nonzero. So yes, 4 is the answer.

Hmm. But maybe I need to check if there's a different interpretation.

Wait, maybe the question is trickier? Like, perhaps when they say "to the right of the decimal point", they mean excluding trailing zeros?

<prompt_2.1> Consider if trailing zeros affect the interpretation of the question and confirm the answer. </prompt_2.1>

But just to make sure, perhaps there's another method to calculate this without converting to decimal? Maybe by manipulating the fraction to find the last nonzero digit.

<prompt_2.2> Explore an alternative method to find the last nonzero digit of 137/500 without converting to decimal. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
But in 0.274, there are no trailing zeros. So the last digit is 4, which is nonzero. So 4 is the answer. 

But let me think again. Maybe the question is from a competition or somewhere where they sometimes have these problems where you have to be careful. For example, sometimes when you have a fraction that converts to a terminating decimal, but maybe if there are more digits when considering more decimal places? But in this case, 137/500 is exactly 0.274, so there's no more digits. So the last nonzero digit is 4.
</execution_2.1>

<execution_2.2>
Let me try that approach to confirm.

So, 137 divided by 500. Let's express this as a decimal. Since 500 is 5*10^2, so 1/500 is 1/(5*100) = (1/5)/100 = 0.2/100 = 0.002. Therefore, 137*0.002 = 0.274. Yep, same result. So that's correct.

Alternatively, if we didn't want to convert to decimal, perhaps we can find the last nonzero digit by considering the fraction in its reduced form and then looking at the decimal expansion.

But since 137 and 500 are coprime? Let's check. 137 is a prime number, right? 137 divided by 2? No. 3? 1+3+7=11, not divisible by 3. 5? Ends with 7, so no. 7? 137/7 is about 19.57, not integer. 11? 137/11 is 12.45, nope. So 137 is prime. 500 is 2^2*5^3. So 137 and 500 share no common factors. Therefore, 137/500 is in simplest terms.

So, when converting to a decimal, since the denominator is 2^2*5^3, to make it a denominator of 10^n, we need to multiply numerator and denominator by 2^(n-3) if n >=3. Wait, in this case, 500 is 2^2*5^3. To convert to a denominator of 10^n, which is 2^n*5^n, we need to have equal exponents for 2 and 5. So since there are 2 2s and 3 5s in the denominator, we need to multiply numerator and denominator by 2^(1) to make the denominator have 2^3*5^3 = 10^3. Therefore, multiplying numerator and denominator by 2 gives 274/1000, which is 0.274. So that's the decimal expansion.

Therefore, the last nonzero digit is 4. So, seems straightforward.

But maybe the problem is more complex? For example, if the decimal expansion was longer, and we had to find the last nonzero digit in a more complicated decimal? Let's see. For example, if the decimal was 0.274444444...., then the last nonzero digit would be 4 as well. But in this case, the decimal terminates at 0.274. So there's no issue with repeating decimals or anything. So 0.274 is exact.

Therefore, the last nonzero digit after the decimal is 4.
</execution_2.2>
</Execution_2>

<Final_answer>
}\), the last nonzero digit to the right of the decimal point is $\boxed{4}$.
</Final_answer>
</think>