Okay, so I need to find the last nonzero digit to the right of the decimal point in the decimal expansion of 137 divided by 500. Hmm, let me think about how to approach this. 

First, I remember that when you divide a number by another, the decimal expansion can be found by long division. Alternatively, maybe converting the fraction into a decimal by manipulating the denominator to make it a power of 10. Let me try that approach since it might be straightforward.

The denominator here is 500. If I can convert 500 into a power of 10, then the numerator will be scaled accordingly, and the decimal should be easier to read. Powers of 10 are 10, 100, 1000, etc. 500 is 5 times 100, so maybe if I multiply both numerator and denominator by 2 to make the denominator 1000, which is a power of 10. Let's check:

Multiplying numerator and denominator by 2:
137 * 2 = 274
500 * 2 = 1000
So, 137/500 = 274/1000. 

Converting 274/1000 to a decimal is straightforward. Since the denominator is 1000, the decimal will have three places. The numerator is 274, so moving the decimal three places to the left:

274 divided by 1000 is 0.274.

Wait, so the decimal expansion is 0.274. The digits after the decimal point are 2, 7, 4. The last nonzero digit here is 4. So is the answer 4? But let me verify to make sure I didn't make a mistake.

Alternatively, maybe I should perform the long division of 137 divided by 500 to confirm. Let's try that.

Dividing 137 by 500:

500 goes into 137 zero times. So we write 0. and then proceed to divide 1370 by 500. 500 goes into 1370 two times (since 2*500=1000). Subtract 1000 from 1370, we get 370. Bring down a zero, making it 3700. 500 goes into 3700 seven times (7*500=3500). Subtract 3500 from 3700, we get 200. Bring down another zero, making it 2000. 500 goes into 2000 four times (4*500=2000). Subtract 2000 from 2000, we get 0. So the division stops here. 

So the decimal expansion is 0.274, same as before. So the digits after the decimal are 2, 7, 4. The last nonzero digit is indeed 4. Therefore, the answer should be 4.

Wait, but just to make sure I didn't skip any steps or make a mistake in the multiplication earlier. Let me check:

Original fraction: 137/500.

Multiply numerator and denominator by 2: 137*2=274, 500*2=1000. So 274/1000=0.274. Correct. So that seems right.

Alternatively, maybe there's a different way to compute the decimal expansion without converting to a denominator of 1000. Let me try another method, like multiplying numerator by 2 to make the denominator 1000, but that's the same as before. Alternatively, using the fact that 500 is 5*100, so 1/500 = 1/(5*100) = (1/5)*(1/100) = 0.2*0.01 = 0.002. Then 137*0.002 = 0.274. Yep, same result.

So all methods lead to 0.274. Therefore, the last nonzero digit after the decimal is 4. 

But just to be thorough, let me check the long division again step by step.

Dividing 137 by 500:

500 | 137.000...

500 doesn't go into 137, so we write 0. and proceed to divide 1370 by 500.

500 goes into 1370 two times (2*500=1000). Subtract 1000 from 1370: 1370-1000=370.

Bring down the next 0, making it 3700.

500 goes into 3700 seven times (7*500=3500). Subtract 3500 from 3700: 3700-3500=200.

Bring down the next 0, making it 2000.

500 goes into 2000 four times (4*500=2000). Subtract 2000 from 2000: 2000-2000=0.

No remainder, so the decimal stops here. So the quotient is 0.274. 

Yes, that's correct. So the decimal expansion is 0.274. The digits after the decimal are 2, 7, 4. The last nonzero digit is 4. 

Therefore, the answer should be 4. 

I think that's solid. All approaches confirm the same result. So I can be confident that the last nonzero digit to the right of the decimal point in the decimal expansion of 137/500 is 4.
Therefore, the last nonzero digit to the right of the decimal point in the decimal expansion of $\frac{137}{500}$ is $\boxed{4}$.
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