Okay, let's tackle this problem step by step. The question is asking for the sum of the digits in the terminating decimal representation of the fraction 4321 divided by (5^7 times 2^8). Hmm, terminating decimals... right, a fraction will have a terminating decimal if the denominator, after simplifying the fraction, has no prime factors other than 2 and 5. So, since the denominator here is already 5^7 times 2^8, that's good. It means the decimal will terminate. Now, the task is to find the sum of the digits in that decimal representation.

First, I need to figure out how to convert this fraction into its decimal form. Since the denominator has both 2s and 5s, the decimal will terminate. The number of decimal places will be the maximum of the exponents of 2 and 5 in the denominator. Wait, let me recall. When converting a fraction to a decimal, especially when the denominator is a product of powers of 2 and 5, the number of decimal places is determined by the highest power between 2 and 5. So here, the denominator is 5^7 * 2^8. The exponents are 7 and 8. The higher one is 8, so the decimal should terminate after 8 digits. But maybe I need to check that.

Alternatively, another approach is to adjust the fraction so that the denominator becomes a power of 10. Since 10 is 2*5, to make the denominator a power of 10, we need the same number of 2s and 5s. In this case, the denominator has 7 5s and 8 2s. To balance them, we can multiply numerator and denominator by 5 to make the number of 5s equal to 8. Let's see:

Original fraction: 4321 / (5^7 * 2^8)

Multiply numerator and denominator by 5 to get:

(4321 * 5) / (5^8 * 2^8) = (4321 * 5) / (10^8)

So the denominator is now 10^8, which is 1 followed by 8 zeros. The numerator becomes 4321 * 5. Let me compute that:

4321 * 5. Let's do the multiplication:

4321 * 5: 4000*5 = 20000, 300*5=1500, 20*5=100, 1*5=5. Adding them up: 20000 + 1500 = 21500, 21500 + 100 = 21600, 21600 + 5 = 21605. So the numerator is 21605.

Therefore, the fraction becomes 21605 / 10^8. Which is 0.00021605. Wait, let me verify that. 10^8 is 100,000,000. So 21605 divided by 100,000,000. To convert that to a decimal, move the decimal point 8 places to the left. Starting from 21605., moving 8 places left: 0.00021605. Let's count the decimal places: the first digit after the decimal is the first place. So 21605 is 5 digits, moving 8 places left would result in 8 - 5 = 3 zeros before the 21605. Wait, no. Wait, 21605 divided by 100,000,000 is 0.00021605. Let's check:

10^8 is 100,000,000. So 21605 divided by 100,000,000. Let's write 21605 as 21605.0. Divided by 100,000,000, which is 1 followed by 8 zeros. So moving the decimal point 8 places to the left:

Starting from 21605.0:

1 place: 2160.5
2 places: 216.05
3 places: 21.605
4 places: 2.1605
5 places: 0.21605
6 places: 0.021605
7 places: 0.0021605
8 places: 0.00021605

Yes, that's correct. So the decimal is 0.00021605. Now, the question is to find the sum of the digits in this decimal representation. Let's write out the decimal: 0.00021605. The digits after the decimal point are 0, 0, 0, 2, 1, 6, 0, 5. So the digits are 0, 0, 0, 2, 1, 6, 0, 5. Wait, is that correct? Let me count the decimal places:

First digit after decimal: 0 (1st place)
Second: 0 (2nd)
Third: 0 (3rd)
Fourth: 2 (4th)
Fifth: 1 (5th)
Sixth: 6 (6th)
Seventh: 0 (7th)
Eighth: 5 (8th)

So the digits are indeed 0, 0, 0, 2, 1, 6, 0, 5. Now, summing these digits: 0 + 0 + 0 + 2 + 1 + 6 + 0 + 5. Let's compute that:

0 + 0 = 0; 0 + 0 = 0; 0 + 2 = 2; 2 + 1 = 3; 3 + 6 = 9; 9 + 0 = 9; 9 + 5 = 14. So the sum is 14. Hmm, but wait, I need to confirm if the decimal representation is correct because sometimes when multiplying by 5, there might be a carryover or something. Let me check the multiplication again.

Original numerator: 4321 * 5. Let's compute 4321 * 5 step by step:

4000 * 5 = 20,000
300 * 5 = 1,500
20 * 5 = 100
1 * 5 = 5
Adding them: 20,000 + 1,500 = 21,500; 21,500 + 100 = 21,600; 21,600 + 5 = 21,605. So that's correct. So 4321 * 5 is indeed 21,605. Then divided by 10^8 is 0.00021605. So the decimal is correct. Then sum of digits: 0+0+0+2+1+6+0+5=14. So the answer should be 14. But wait, let me check again. Maybe I missed a digit or added incorrectly.

Digits: 0, 0, 0, 2, 1, 6, 0, 5. So adding them:

First three zeros: 0.

Then 2, 1, 6, 0, 5. So 2+1=3, 3+6=9, 9+0=9, 9+5=14. Yes, that's correct. So the sum is 14.

But just to be thorough, maybe there's another way to approach this problem without converting to decimal. Let me think. The original fraction is 4321/(5^7 * 2^8). Since the denominator is 5^7 * 2^8, and we can express this as (5^7 * 2^7) * 2^1 = 10^7 * 2. So the denominator can be written as 10^7 * 2. Therefore, the fraction becomes 4321/(10^7 * 2) = (4321/2)/10^7. Let's compute 4321 divided by 2. 4321 / 2 = 2160.5. Then divide by 10^7, which is moving the decimal point 7 places to the left. So 2160.5 / 10^7 = 0.00021605. Which is the same result as before. So the decimal is indeed 0.00021605, sum of digits 14. So that seems consistent.

Alternatively, perhaps there's a way to compute the decimal without converting to 10^8. Let's see. The original denominator is 5^7 * 2^8. Let's factor that as 5^7 * 2^7 * 2^1 = 10^7 * 2. So the denominator is 10^7 * 2. Therefore, the fraction is 4321/(10^7 * 2). Which is the same as (4321/2)/10^7. As above, 4321 divided by 2 is 2160.5, then divided by 10^7 gives 0.00021605. So same result.

Alternatively, maybe we can compute the decimal representation by adjusting the fraction. Let's see. Since 4321 is the numerator, and the denominator is 5^7 * 2^8. Let's write this as 4321/(2^8 * 5^7) = (4321 / 2^7) / (2 * 5^7) = (4321 / 128) / (10^7). Wait, 2^7 is 128. So 4321 divided by 128. Let's compute that:

4321 ÷ 128. Let's do the division:

128 * 30 = 3840. 4321 - 3840 = 481.

128 * 3 = 384. 481 - 384 = 97.

So 4321 ÷ 128 = 33. remainder 97. So 33.97/128? Wait, no. Wait, 4321 divided by 128 is 33 with a remainder of 97. So that's 33 + 97/128. Then divide by 10^7: (33 + 97/128)/10^7 = 33/10^7 + 97/(128*10^7). Let's compute 33/10^7 = 0.0000033. And 97/(128*10^7). Let's compute 97/128 first. 97 ÷ 128 = 0.7578125 (since 128 * 0.75 = 96, so 0.75 + 0.0078125 = 0.7578125). Then divided by 10^7: 0.7578125 × 10^-7 = 0.00000007578125. Therefore, adding 0.0000033 + 0.00000007578125 ≈ 0.00000337578125. Wait, but that's conflicting with the previous result. Wait, this can't be. There must be a mistake here.

Wait, maybe I made a mistake in the division steps. Let's check. 4321 divided by 128. Let's do it step by step.

128 x 30 = 3840. 4321 - 3840 = 481.

128 x 3 = 384. 481 - 384 = 97. So 4321 ÷ 128 = 33 with remainder 97, so 33 + 97/128. Therefore, 4321/(128 * 10^7) = (33 + 97/128)/10^7. Which is 33/10^7 + 97/(128*10^7). 33/10^7 is 0.0000033, and 97/(128*10^7) is (97/128)/10^7. 97/128 is 0.7578125, so divided by 10^7 is 0.00000007578125. Adding those together: 0.0000033 + 0.00000007578125 = 0.00000337578125. Wait, but this contradicts the previous result of 0.00021605. So which is correct?

Wait, 4321/(5^7 * 2^8) is equal to 4321/(10^7 * 2) = 4321/(2 * 10^7). So 4321 ÷ 2 = 2160.5, then ÷ 10^7: 0.00021605. So that's correct. But in the other approach, dividing by 128 first, then by 10^7, I get a different result. So there must be an error in the second approach.

Wait, wait. Let's see. 4321/(5^7 * 2^8) = 4321/(2^8 * 5^7). Let's factor 2^7 from the denominator: 2^7 * 2^1 * 5^7 = 10^7 * 2. Therefore, 4321/(10^7 * 2). So that's 4321/(2 * 10^7). Which is 4321 divided by 2, which is 2160.5, divided by 10^7: 0.00021605. So that's correct. So the second approach must have a mistake. Let me check:

I wrote 4321/(128 * 10^7). But 128 is 2^7, so 2^7 * 10^7 is 2^7 * (2^7 * 5^7) = 2^14 * 5^7. Wait, no. Wait, 10^7 is 2^7 * 5^7. Therefore, 2^7 * 10^7 is 2^7 * 2^7 * 5^7 = 2^14 * 5^7. But the original denominator is 2^8 * 5^7. Therefore, 2^8 * 5^7 = 2^(8) * 5^(7). So the denominator is 2^8 * 5^7. So when I wrote 4321/(128 * 10^7), that is 4321/(2^7 * 10^7) = 4321/(2^7 * 2^7 * 5^7) = 4321/(2^14 * 5^7). But the original fraction is 4321/(2^8 * 5^7). So that's different. Therefore, the mistake was in the second approach: I incorrectly represented the denominator as 2^7 * 10^7 instead of 2^8 * 5^7. Therefore, that approach was wrong. Therefore, the first approach where we adjust the fraction to have denominator 10^8 is correct, leading to the decimal 0.00021605.

Therefore, the sum of the digits is 14. So the answer is 14.

But just to confirm, let's check another way. Let's compute the decimal expansion manually. 4321 divided by (5^7 * 2^8). Let's compute the decimal step by step.

First, note that 5^7 = 78125, and 2^8 = 256. So denominator is 78125 * 256. Let's compute that:

78125 * 256. Let's compute 78125 * 200 = 15,625,000. 78125 * 50 = 3,906,250. 78125 * 6 = 468,750. Adding those together: 15,625,000 + 3,906,250 = 19,531,250; 19,531,250 + 468,750 = 20,000,000. Wait, really? 78125 * 256 = 20,000,000? Let me check:

78125 * 256. 78125 * 200 = 15,625,000. 78125 * 56 = 4,375,000. 15,625,000 + 4,375,000 = 20,000,000. Yes, that's correct. So denominator is 20,000,000. Then the fraction is 4321 / 20,000,000. To convert that to decimal, divide 4321 by 20,000,000.

Alternatively, 4321 / 20,000,000 = 0.00021605. Yes, exactly as before. So 0.00021605. So the decimal is 0.00021605. So the digits after the decimal are 0,0,0,2,1,6,0,5. Sum is 0+0+0+2+1+6+0+5=14. So that's correct.

Therefore, the answer is 14.
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