Explanation:
Okay, so I need to figure out how many digits repeat in the decimal expansion of 31 divided by 11111. Let me start by recalling that when you divide a number by another, if the denominator has prime factors other than 2 and 5, the decimal will repeat. Since 11111 is the denominator here, I should check its prime factors to understand the repeating pattern.

First, let me try dividing 31 by 11111 directly. But maybe there's a smarter way than long division. Let me think. The denominator is 11111. Hmm, 11111... that number looks familiar. Wait, 11111 is 10^5 - 1 over 9? Let me check: 10^5 is 100000, so 100000 - 1 is 99999. Divided by 9 is 11111. Yes! So 11111 = (10^5 - 1)/9. Therefore, 1/11111 is 9/(10^5 - 1). So, 1/11111 is equal to 9/99999, which is 0.000090000900009..., repeating every 5 digits? Wait, but 9/99999 is 0.\overline{00009}? Let me verify that.

If I take 1/99999, that's 0.\overline{00001}, right? Because 1/99999 = 0.000010000100001..., repeating every 5 digits. Then multiplying by 9 would give 0.000090000900009..., so the repeating part is "00009", which is 5 digits. Therefore, 1/11111 is 0.\overline{00009}, so the repeating decimal has 5 digits. Therefore, 31/11111 would be 31 times that decimal. So, 31 * 0.\overline{00009}.

But how does multiplying by 31 affect the repeating decimal? Let me think. If I have 0.\overline{00009}, multiplying by 31 would give me a decimal that's 31 times that. Let me try to compute that. Alternatively, maybe there's a pattern here. Let me think about how multiplying by 31 would affect the repeating sequence.

Alternatively, maybe I can perform the division directly. Let me try dividing 31 by 11111. Since 31 is less than 11111, the decimal will start with 0. Let me write this out as a long division problem.

So, 31 divided by 11111. Let me add decimal places and zeros to 31. So, 31.000000...

First, 11111 goes into 310 zero times. Then 3100, still zero. 31000, still zero. 310000 divided by 11111. Let me calculate how many times 11111 fits into 310000.

Let me compute 11111 * 2 = 22222. 11111 * 3 = 33333. But 33333 is larger than 310000? Wait, no. Wait, 11111 * 2 = 22222. 11111 * 27 = ?

Wait, maybe I need to compute 310000 divided by 11111. Let me approximate. 11111 * 2 = 22222. 11111 * 28 = 11111 * 27 + 11111. Let me compute 11111 * 27. 11111 * 20 = 222220, 11111 * 7 = 77777. Adding them together: 222220 + 77777 = 299,997. Then adding another 11111 gives 299,997 + 11,111 = 311,108. But 311,108 is larger than 310,000. So 11111 * 27 = 299,997. So 299,997 is less than 310,000. The difference is 310,000 - 299,997 = 10,003. So, 11111 goes into 310,000 27 times with a remainder of 10,003. Therefore, the first digit after the decimal is 0. Then, after the decimal, we have 0. So, first, we have 0. Then, after the decimal, we have 0. Then, we bring down zeros. Wait, maybe I need to track this more carefully.

Wait, let me start over. Dividing 31 by 11111:

31 ÷ 11111 = 0. So, we write 0. and then proceed with the decimal.

Multiply 31 by 10: 310. Still less than 11111. Next digit 0.

Multiply by 10 again: 3100. Still less. Next digit 0.

Multiply by 10 again: 31000. Still less. Next digit 0.

Multiply by 10 again: 310000. Now, divide 310000 by 11111. As above, 11111 * 27 = 299,997. So, 27. Then subtract 299,997 from 310,000: 310,000 - 299,997 = 10,003. Bring down a zero: 100,030.

Now divide 100,030 by 11111. Let me see how many times 11111 fits into 100,030. Let me compute 11111 * 9 = 99,999. That's very close. So 9 times. Subtract 99,999 from 100,030: 100,030 - 99,999 = 31. Bring down a zero: 310.

Now, we have 310 again. Wait, this is the same as the original numerator. So, the decimal repeats here. So, the decimal expansion is 0.000279... and then repeats?

Wait, let me track the digits. Let me write down the steps:

First, after the decimal, we had four zeros (from multiplying by 10 four times), then we had 27, then 9, then we get back to 310, which is where we started. So, the decimal expansion is 0.000279000279000279..., repeating every 6 digits? Wait, let me check.

Wait, let me retrace:

Starting with 31.000000...

First division steps:

- 31 divided by 11111: 0.
- 310 divided by 11111: 0.
- 3100 divided by 11111: 0.
- 31000 divided by 11111: 0.
- 310000 divided by 11111: 27 (as 27*11111=299997), remainder 10003.
- Bring down a 0: 100030 divided by 11111: 9 (9*11111=99999), remainder 31.
- Bring down a 0: 310 divided by 11111: 0.
- Bring down a 0: 3100 divided by 11111: 0.
- Bring down a 0: 31000 divided by 11111: 0.
- Bring down a 0: 310000 divided by 11111: 27 again...

Wait, so after the first 27 and 9, we get back to 310, which is the same as the original numerator. Therefore, the decimal expansion is 0.000279000279000279..., so the repeating part is "000279"? Wait, but let me check the digits.

Wait, let me write down the decimal digits step by step:

After the decimal point, the first four digits are 0s (from the first four divisions). Then, when we divided 310000 by 11111, we got 27, so the fifth digit is 2, sixth is 7. Then, after subtracting, we had remainder 10003, then bringing down a zero gives 100030. Dividing that by 11111 gives 9, so the seventh digit is 9. Then remainder 31, bringing down a zero gives 310. Then, we have to divide 310 by 11111, which is 0, so the eighth digit is 0. Then bring down a zero to get 3100, which is 0, ninth digit 0. Bring down a zero to get 31000, which is 0, tenth digit 0. Bring down a zero to get 310000 again, which is 27, so the eleventh digit is 2, twelfth is 7, and so on.

So the decimal expansion is 0.000279000279000279..., so the repeating part is "000279", which is 6 digits. Wait, but let me check if that's correct. Let me count the digits between the repeating parts. From the first 0.000279... then the next digits are 000279... So, the repeating cycle is 000279, which is 6 digits. Therefore, the answer would be 6?

But earlier, I thought that 1/11111 is 0.\overline{00009}, which is 5 digits. But when multiplying by 31, the decimal becomes 0.\overline{000279}? Wait, but 0.\overline{000279} would be 000279 repeating. Let me check if that's correct.

Alternatively, maybe there's a mistake here. Let me verify with another approach. Since 1/11111 = 0.\overline{00009}, then multiplying by 31 would give 31 * 0.\overline{00009} = 0.\overline{00009 * 31}?

Wait, but multiplying a repeating decimal by a number doesn't necessarily just multiply the repeating part. For example, 0.\overline{abc} * n is not necessarily 0.\overline{abc * n} unless n is a single digit. So maybe my previous approach is flawed.

Alternatively, let me think of 31/11111 as 31/(10^5 - 1)/9. Wait, earlier I said 11111 = (10^5 - 1)/9. Therefore, 31/11111 = 31 * 9 / (10^5 - 1) = 279 / 99999. Therefore, 279 divided by 99999. Which is 0.\overline{00279}? Wait, 279/99999. Let me see. Since 279/99999 = 0.002790027900279..., so the repeating part is "00279", which is 5 digits? Wait, but 279 divided by 99999. Let me check:

If I have 279 divided by 99999, then since 99999 is 10^5 - 1, so 279/(10^5 - 1) = 0.\overline{00279}. Wait, but how many digits? Let me think. If I have a fraction a/(10^n - 1), then the decimal repeats every n digits. So here, 279/(10^5 - 1) would have a repeating cycle of 5 digits. But 279 is a 3-digit number. So, how does that work?

Wait, for example, 1/99999 = 0.\overline{00001}, which is 5 digits. Similarly, 279/99999 would be 0.\overline{00279}, which is 5 digits. But 00279 is 5 digits. So, the decimal expansion is 0.002790027900279..., so the repeating part is 00279, which is 5 digits. But earlier, when I did the long division, I thought it was 0.000279000279..., which is 6 digits. There's a contradiction here. Which one is correct?

Wait, let me check with actual calculation. Let me compute 279 divided by 99999. Let me do that division. 279 ÷ 99999. Since 279 is less than 99999, it's 0. Then, 279.000000...

Multiply by 10: 2790. Still less than 99999. Next digit 0.

Multiply by 10: 27900. Still less. Next digit 0.

Multiply by 10: 279000. Still less. Next digit 0.

Multiply by 10: 2790000. Now divide by 99999. Let me see how many times 99999 fits into 2790000.

Well, 99999 * 27 = 2699973. Let me check 99999 * 27:

99999 * 20 = 1,999,980

99999 * 7 = 699,993

Adding them: 1,999,980 + 699,993 = 2,699,973. Then, 2,699,973. Subtract that from 2,790,000: 2,790,000 - 2,699,973 = 90,027. Bring down a zero: 900,270.

Divide 900,270 by 99,999. Let me see, 99,999 * 9 = 899,991. Subtract that from 900,270: 900,270 - 899,991 = 279. Bring down a zero: 2790. Now we are back to where we started. So the decimal expansion is 0.000279000279..., so the repeating part is "000279", which is 6 digits. Wait, but according to the previous logic, it should be 5 digits. What's going on here?

Wait, maybe my assumption that 279/99999 is 0.\overline{00279} is wrong. Let me check with another approach. Let me note that 279/99999 = (279/9)/ (11111) = 31/11111. Which is the original fraction. So, if I have 279/99999, which is equal to 0.000279000279..., then the repeating cycle is 6 digits. But why does the formula a/(10^n - 1) have a repeating cycle of n digits? Maybe because if the numerator has fewer digits, it's padded with leading zeros. For example, 279/99999 is equivalent to 00279/99999, so the decimal is 0.0027900279..., but that would be 5 digits. Wait, but in reality, when I did the division, I got 0.000279000279..., which is 6 digits. There's a contradiction here. Which is correct?

Wait, let me check with a calculator. Let me compute 31 divided by 11111. Let me approximate. 11111 * 0.00279 = ?

Well, 11111 * 0.002 = 22.222

11111 * 0.00079 = ?

First, 11111 * 0.0007 = 7.7777

11111 * 0.00009 = 0.99999

So total is 7.7777 + 0.99999 ≈ 8.77769

So total 0.00279 * 11111 ≈ 22.222 + 8.77769 ≈ 30.99969, which is approximately 31. So that suggests that 31/11111 ≈ 0.0027900279..., so the decimal is 0.0027900279..., which would be repeating every 5 digits? Wait, but according to the long division, it was 0.000279000279... which is different. Wait, there's confusion here. Let me check again.

Wait, if 0.0027900279... is the decimal, then that would be 0.00279 repeating, which is 5 digits. But according to the long division, I had 0.000279000279..., which is 0.000279 repeating, which is 6 digits. Which is correct?

Wait, let me check with another method. Let me compute 31/11111 using another approach. Let me note that 1/11111 = 0.\overline{00009} as I thought earlier. Then 31 * 0.\overline{00009} = ?

Multiplying a repeating decimal by an integer. Let me think. If I have 0.\overline{00009} = 0.000090000900009..., then multiplying by 31:

Let me write it as 0.000090000900009... * 31.

Let me compute 0.00009 * 31 = 0.00009 * 30 + 0.00009 * 1 = 0.0027 + 0.00009 = 0.00279.

Then, the rest of the decimal is 0.0000000000900009... * 31, which would be 0.000000000279..., so adding that to 0.00279 gives 0.002790000279..., so the decimal is 0.002790000279..., which is 0.00279 repeating? Wait, but the decimal is 0.002790000279..., so the repeating part is "0000279"? Wait, no. Wait, if you have 0.002790000279..., then the repeating part is "0000279"? Wait, but how?

Wait, maybe I need to think of the decimal expansion as follows. If 1/11111 = 0.\overline{00009}, then multiplying by 31 gives 0.\overline{00009 * 31}?

But 00009 * 31 is 279. So, 0.\overline{279}? But that would be 0.279279279..., but that's not matching with the previous calculation. Wait, there's confusion here. Let me think again.

If I have 0.\overline{abc}, then multiplying by n gives a decimal that depends on the value of n. For example, multiplying by 31 would not just multiply the repeating block. However, if the repeating block is of length k, then multiplying by n could result in a different repeating block. However, in this case, since 1/11111 = 0.\overline{00009}, which is 5 digits, multiplying by 31 would give a decimal that is 31 * 0.\overline{00009} = 0.\overline{00009 * 31} only if 00009 * 31 doesn't cause a carryover that affects the repeating block. But 00009 * 31 is 279, which is 3 digits. So, if we have 0.\overline{00009} * 31 = 0.\overline{279}?

But that would be 0.279279279..., but that contradicts the previous calculation where we had 0.00279... So, clearly, there's a mistake here. Where is the error?

Ah! Wait, 0.\overline{00009} is not 0.0000900009..., but rather 0.0000900009... which is 0.00009 repeating. So, multiplying by 31 would be 0.00009 * 31 = 0.00279, and then the rest of the decimal would be 0.00000000009 * 31 = 0.000000000279, etc. So, the decimal expansion is 0.0027900002790000279..., which is 0.002790000279..., so the repeating part is "0000279"? Wait, but how?

Alternatively, maybe the decimal expansion is 0.002790000279..., which is 0.00279 repeating? But that would require that the decimal is 0.002790000279..., which is not a simple repeating of 00279. Wait, maybe the repeating cycle is longer. Let me check with the long division result.

Earlier, when I did the long division, I found that the decimal expansion is 0.000279000279..., which is 0.000279 repeating. But according to the multiplication approach, it's 0.00279... So which is correct?

Wait, let me check with actual decimal multiplication. Let me compute 31/11111 numerically. Let me approximate 1/11111 ≈ 0.00009. Then 31 * 0.00009 = 0.00279. But if I compute 31/11111, let me do a better approximation. Let me compute 1/11111 ≈ 0.0000900009... So multiplying by 31 gives 0.0000900009... * 31 = 0.0027900279..., which is 0.0027900279..., so the decimal is 0.0027900279..., which is 0.00279 repeating? Wait, but the decimal is 0.0027900279..., so the repeating part is "00279"? Wait, but how?

Wait, if I write 0.002790027900279..., then the repeating part is "00279", which is 5 digits. But in the long division, I had 0.000279000279..., which is 0.000279 repeating, which is 6 digits. There's a contradiction here. Which one is correct?

Wait, let me check with another method. Let me compute 31 divided by 11111 using a calculator. Since I don't have a calculator here, let me try to compute it more carefully.

Let me consider that 1/11111 ≈ 0.000090000900009...

So multiplying by 31:

First, 0.00009 * 31 = 0.00279

Then, the next part is 0.00000000009 * 31 = 0.000000000279

Adding these together: 0.00279 + 0.000000000279 = 0.002790000279...

So, the decimal expansion is 0.002790000279..., which is 0.002790000279..., so the repeating part is "0000279"? Wait, but how?

Wait, if I write it out:

0.0027900002790000279...

So, after the decimal, the digits are:

0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9,...

So, the repeating part is "0000279", which is 7 digits? Wait, but that seems inconsistent. Wait, let me count:

Starting from the decimal point:

Position 1: 0

Position 2: 0

Position 3: 2

Position 4: 7

Position 5: 9

Position 6: 0

Position 7: 0

Position 8: 0

Position 9: 0

Position 10: 2

Position 11: 7

Position 12: 9

Position 13: 0

Position 14: 0

Position 15: 0

Position 16: 0

Position 17: 2

...

So, the pattern is "00279" followed by four zeros? Wait, no. Wait, from position 1-5: 0,0,2,7,9; then positions 6-9: 0,0,0,0; then positions 10-14: 2,7,9,0,0; then positions 15-19: 0,0,0,0,2... Wait, this seems inconsistent. Maybe my previous assumption is wrong.

Alternatively, maybe there's a mistake in the long division. Let me go back to the long division steps.

Earlier, I had:

After dividing 31 by 11111, I had:

- 31.000000...

First four divisions gave 0.0000...

Then, dividing 310000 by 11111 gave 27, remainder 10003.

Then, bringing down a zero: 100030 divided by 11111 is 9, remainder 31.

Then, bringing down a zero: 310 divided by 11111 is 0, remainder 310.

Then, bringing down a zero: 3100 divided by 11111 is 0, remainder 3100.

Then, bringing down a zero: 31000 divided by 11111 is 0, remainder 31000.

Then, bringing down a zero: 310000 divided by 11111 is 27, remainder 10003...

So, the decimal digits obtained are:

After the decimal: 0, 0, 0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9,...

So, the decimal expansion is 0.000027900002790000279..., so the repeating part is "0000279", which is 7 digits? Wait, but how?

Wait, let me count the digits between the repeating parts. From the first 0.0000279... then the next digits are 0000279... So, the repeating cycle is "0000279", which is 7 digits? But that seems inconsistent with the previous thought that it should be 5 digits. What's going on here?

Alternatively, maybe there's a mistake in the long division steps. Let me check again:

Starting with 31 divided by 11111:

- 31.000000...

First, 11111 goes into 31 zero times. Write 0.

Multiply by 10: 310. Still 0. Write 0.

Multiply by 10: 3100. Still 0. Write 0.

Multiply by 10: 31000. Still 0. Write 0.

Multiply by 10: 310000. Now divide by 11111.

As before, 11111 * 27 = 299,997. Subtract from 310,000: 310,000 - 299,997 = 10,003. Write down 27. So, the decimal so far is 0.000027...

Bring down a 0: 100,030. Divide by 11111. 11111 * 9 = 99,999. Subtract: 100,030 - 99,999 = 31. Write down 9. Decimal is 0.0000279...

Bring down a 0: 310. Divide by 11111: 0. Write down 0. Decimal is 0.00002790...

Bring down a 0: 3100. Divide by 11111: 0. Write down 0. Decimal is 0.000027900...

Bring down a 0: 31000. Divide by 11111: 0. Write down 0. Decimal is 0.0000279000...

Bring down a 0: 310000. Divide by 11111: 27. Write down 27. Decimal is 0.0000279000027...

So, the repeating cycle is "0000279", which is 7 digits? Wait, but the digits after the decimal are:

0.000027900002790000279...

So, the repeating part is "0000279", which is 7 digits? Wait, but how? Because after the first 0.0000279, we have 0000279 again. Wait, but the first "0000279" is positions 1-7: 0,0,0,0,2,7,9. Then the next digits are 0,0,0,0,2,7,9... So, the repeating cycle is 7 digits? But that seems odd. However, according to the long division, that's what it is.

But earlier, when I thought of 279/99999, I thought it would be 0.\overline{00279}, which is 5 digits. But according to the long division, it's 0.\overline{0000279}, which is 7 digits. There's a contradiction here. Where is the mistake?

Wait, let me check 279 divided by 99999. If I write 279/99999, that's equal to 0.002790027900279..., which is 0.00279 repeating every 5 digits? Wait, but how?

Wait, 279/99999 = 0.002790027900279... So, the decimal is 0.0027900279..., which is 0.00279 repeating? But the repeating part is "00279", which is 5 digits. However, in the long division, I have 0.00002790000279..., which is different. So, which one is correct?

Wait, there must be a miscalculation in one of the methods. Let me check with fractions. Let me compute 279/99999. Simplify the fraction. Let's see if 279 and 99999 have common factors.

Divide numerator and denominator by 3: 279 ÷ 3 = 93; 99999 ÷ 3 = 33333. So, 93/33333. Divide by 3 again: 93 ÷ 3 = 31; 33333 ÷ 3 = 11111. So, 31/11111, which is the original fraction. So, 279/99999 = 31/11111. Therefore, both methods are consistent. So, if 279/99999 = 0.0027900279..., then why does the long division give me 0.00002790000279...?

Wait, there must be an error in my long division steps. Let me check again.

Wait, when I did the long division earlier, I had:

After dividing 31 by 11111, I had:

- 31.000000...

First four divisions gave 0.0000...

Then, dividing 310000 by 11111 gave 27, remainder 10003. So, the decimal digits so far are 0.000027...

Then, bringing down a zero: 100030 divided by 11111 is 9, remainder 31. So, decimal is 0.0000279...

Then, bringing down a zero: 310 divided by 11111 is 0, remainder 310. So, decimal is 0.00002790...

Then, bringing down a zero: 3100 divided by 11111 is 0, remainder 3100. Decimal is 0.000027900...

Then, bringing down a zero: 31000 divided by 11111 is 0, remainder 31000. Decimal is 0.0000279000...

Then, bringing down a zero: 310000 divided by 11111 is 27, remainder 10003. Decimal is 0.0000279000027...

So, the decimal expansion is 0.000027900002790000279..., which is 0.0000279 repeating? Wait, but how?

Wait, if I write this out, it's 0.000027900002790000279..., so the repeating part is "0000279", which is 7 digits. But according to the fraction 279/99999, it should be 0.0027900279..., which is different. Where is the mistake?

Wait, maybe I made a mistake in the long division steps. Let me check the first few digits again.

Starting with 31 divided by 11111:

- 31 ÷ 11111 = 0. So, write 0.
- 31.0 ÷ 11111: 0.0
- 31.00 ÷ 11111: 0.00
- 31.000 ÷ 11111: 0.000
- 31.0000 ÷ 11111: 0.0000
- Now, 31.00000 ÷ 11111: Let's consider 310000 ÷ 11111.

Wait, here's the problem! When I did the long division, I considered 31.00000... as 31.00000, but when I multiply by 10^5, it's 3100000? Wait, no. Wait, let me clarify:

When performing long division, after the decimal point, each step is multiplying by 10. So starting with 31.000000...

First division step: 31 ÷ 11111 = 0, remainder 31. Bring down a 0: 310.

Second division step: 310 ÷ 11111 = 0, remainder 310. Bring down a 0: 3100.

Third division step: 3100 ÷ 11111 = 0, remainder 3100. Bring down a 0: 31000.

Fourth division step: 31000 ÷ 11111 = 0, remainder 31000. Bring down a 0: 310000.

Fifth division step: 310000 ÷ 11111 = 27, remainder 10003. So, the fifth digit after the decimal is 2, sixth is 7. So, decimal is 0.000027...

Then, bring down a 0: 100030 ÷ 11111 = 9, remainder 31. So, seventh digit is 9. Decimal is 0.0000279...

Bring down a 0: 310 ÷ 11111 = 0, remainder 310. Eighth digit is 0. Decimal is 0.00002790...

Bring down a 0: 3100 ÷ 11111 = 0, remainder 3100. Ninth digit is 0. Decimal is 0.000027900...

Bring down a 0: 31000 ÷ 11111 = 0, remainder 31000. Tenth digit is 0. Decimal is 0.0000279000...

Bring down a 0: 310000 ÷ 11111 = 27, remainder 10003. Eleventh digit is 2, twelfth is 7. Decimal is 0.0000279000027...

So, the decimal expansion is 0.000027900002790000279..., which is 0.0000279 repeating every 7 digits? Wait, but how does this relate to 279/99999?

Wait, 279/99999 = 0.002790027900279... So, why is there a discrepancy?

Wait, maybe I made a mistake in the long division. Let me check the value of 0.00002790000279... versus 0.0027900279...

Let me convert 0.00002790000279... to a fraction. Let me denote x = 0.00002790000279...

Multiply x by 10^7 (since the repeating cycle is 7 digits): 10^7 * x = 2790000279... Wait, no. Wait, if the repeating cycle is 7 digits, then multiplying by 10^7 would shift the decimal 7 places. Let me try:

Let me assume the repeating cycle is "0000279", which is 7 digits. So, x = 0.000027900002790000279...

Let me write x = 0.000027900002790000279...

Let me multiply x by 10^7 to shift the decimal 7 places:

10^7 * x = 279.00002790000279...

But the original x is 0.00002790000279...

So, 10^7 * x = 279 + x

Therefore, 10^7 * x = 279 + x

Subtract x: 10^7 * x - x = 279

x(10^7 - 1) = 279

x = 279 / (10^7 - 1) = 279 / 9999999

But earlier, we have x = 31/11111 = 279/99999. But 279/99999 is not equal to 279/9999999. So, this suggests that my assumption about the repeating cycle being 7 digits is wrong. Therefore, there must be an error in the long division steps.

Wait, this is confusing. Let me check with the fraction 279/99999. Let me compute 279 divided by 99999. Let me do this division properly.

Let me write 279 ÷ 99999. Since 279 is less than 99999, it's 0. So, 0. and then proceed.

Multiply by 10: 2790. Still less than 99999. Next digit 0.

Multiply by 10: 27900. Still less. Next digit 0.

Multiply by 10: 279000. Still less. Next digit 0.

Multiply by 10: 2790000. Now divide by 99999.

How many times does 99999 go into 2790000? Let me calculate 99999 * 27 = 2699973. Subtract from 2790000: 2790000 - 2699973 = 90027. Bring down a 0: 900270.

Divide 900270 by 99999. 99999 * 9 = 899991. Subtract: 900270 - 899991 = 279. Bring down a 0: 2790. Now we are back to where we started. So, the decimal expansion is 0.00002790000279..., which is 0.0000279 repeating every 6 digits? Wait, no. Wait, the digits after the decimal are:

0.000027900002790000279...

So, the repeating part is "0000279", which is 7 digits? But according to this division, after the first 0.0000279, we have 0000279 again. Wait, but the remainder after the first division was 279, and then we brought down a zero to get 2790, which is the same as the original numerator. Wait, no. Wait, the original numerator was 279, and after bringing down a zero, it became 2790, which is different. Wait, no, in this case, the original numerator was 279, and after multiplying by 10, it became 2790, then 27900, etc. But when we divided 2790000 by 99999, we got 27 with remainder 90027, then brought down a zero to get 900270, divided by 99999 to get 9 with remainder 279, then brought down a zero to get 2790, which is the same as the original numerator multiplied by 10. Wait, no, the original numerator was 279, and now it's 2790. So, this suggests that the decimal repeats every 6 digits? Wait, let me count the digits:

Starting from 0.0000279...

After the decimal, the digits are:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 0

Position 5: 2

Position 6: 7

Position 7: 9

Position 8: 0

Position 9: 0

Position 10: 0

Position 11: 0

Position 12: 2

Position 13: 7

Position 14: 9

...

So, the repeating cycle is "0000279", which is 7 digits. But according to the fraction, it should be 279/99999 = 0.0027900279..., which is different. Wait, but if I write 0.00002790000279..., that is equal to 0.00002790000279... which is 0.0000279 repeating every 7 digits. But how does this relate to 279/99999?

Wait, let me compute 0.00002790000279... as a fraction. Let me denote x = 0.00002790000279...

Let me multiply x by 10^7 to shift the decimal 7 places: 10^7 * x = 279.00002790000279...

But the original x is 0.00002790000279..., so 10^7 * x = 279 + x

Therefore, 10^7 * x - x = 279

x(10^7 - 1) = 279

x = 279 / (10^7 - 1) = 279 / 9999999

But earlier, we have x = 31/11111 = 279/99999. But 279/99999 is not equal to 279/9999999. So, this suggests that there is a contradiction, which means my long division must have an error.

Wait, but earlier, we saw that 31/11111 = 279/99999. So, if x = 279/99999, then x = 0.0027900279..., which is different from the long division result. So, where is the mistake?

Wait, let me check the value of 279/99999. Let me compute 279 divided by 99999. Let me do this division properly.

Let me write 279 ÷ 99999. Since 279 is less than 99999, it's 0. So, 0. and then proceed.

Multiply by 10: 2790. Still less than 99999. Next digit 0.

Multiply by 10: 27900. Still less. Next digit 0.

Multiply by 10: 279000. Still less. Next digit 0.

Multiply by 10: 2790000. Now divide by 99999.

As before, 99999 * 27 = 2699973. Subtract from 2790000: 2790000 - 2699973 = 90027. Bring down a 0: 900270.

Divide 900270 by 99999: 99999 * 9 = 899991. Subtract: 900270 - 899991 = 279. Bring down a 0: 2790. Now, this is the same as the original numerator multiplied by 10. Wait, original numerator was 279, now it's 2790. So, this suggests that the decimal repeats every 6 digits? Wait, let me count the digits:

After the decimal, the digits are:

0.00002790000279...

Wait, but how? Let me write out the decimal places:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 0

Position 5: 2

Position 6: 7

Position 7: 9

Position 8: 0

Position 9: 0

Position 10: 0

Position 11: 0

Position 12: 2

Position 13: 7

Position 14: 9

...

So, the repeating cycle is "0000279", which is 7 digits. But according to the fraction, it should be 0.0027900279..., which is different. Wait, but if I write 0.00002790000279... as 0.00002790000279..., this is equal to 0.0000279 + 0.0000000000279 + ... which is a geometric series. Let me compute the sum.

Let me denote x = 0.00002790000279...

This can be written as x = 0.0000279 + 0.0000000000279 + 0.0000000000000000279 + ...

This is a geometric series with first term a = 0.0000279 and common ratio r = 10^{-6} (since each term is 10^{-6} times the previous one). So, the sum is a / (1 - r) = 0.0000279 / (1 - 10^{-6}) ≈ 0.0000279 / 0.999999 ≈ 0.00002790000279..., which is consistent with the long division. But how does this relate to 279/99999?

Wait, let me compute 279/99999. Let me do this division properly. Let me write 279 divided by 99999.

Let me note that 279/99999 = (279/9)/ (11111) = 31/11111, which is the original fraction. So, if I compute 31/11111, it should be equal to 279/99999, which is equal to 0.0027900279... or 0.00002790000279...? There's a contradiction here.

Wait, let me compute 279/99999 numerically. Let me approximate it. Let me note that 1/99999 ≈ 0.000010000100001... So, 279 * 0.000010000100001... = 0.002790027900279... So, this is 0.0027900279..., which is 0.00279 repeating every 5 digits. But according to the long division, it's 0.00002790000279..., which is different. So, there must be a mistake in the long division steps.

Wait, let me check the long division again. When I did the long division for 31/11111, I got 0.00002790000279..., but according to the fraction, it should be 0.0027900279... So, where is the error?

Wait, let me check the value of 0.00002790000279... and 0.0027900279... Let me convert both to fractions.

First, 0.00002790000279... = 279/9999999 (as per earlier calculation). Let me compute 279/9999999. Simplify this fraction. Divide numerator and denominator by 3: 93/3333333. Divide by 3 again: 31/1111111. So, 31/1111111. But the original fraction is 31/11111. So, these are different. Therefore, there must be a mistake in the long division.

Wait, this suggests that my long division was incorrect. Let me try again.

Let me perform the long division of 31 divided by 11111 carefully.

Start with 31.000000...

11111 goes into 31 zero times. Write 0.

Bring down a 0: 310. Still 0. Write 0.

Bring down a 0: 3100. Still 0. Write 0.

Bring down a 0: 31000. Still 0. Write 0.

Bring down a 0: 310000. Now, divide 310000 by 11111.

As before, 11111 * 27 = 299997. Subtract from 310000: 310000 - 299997 = 10003. Write 27. So, the decimal is 0.000027...

Bring down a 0: 100030. Divide by 11111. 11111 * 9 = 99999. Subtract: 100030 - 99999 = 31. Write 9. Decimal is 0.0000279...

Bring down a 0: 310. Divide by 11111: 0. Write 0. Decimal is 0.00002790...

Bring down a 0: 3100. Divide by 11111: 0. Write 0. Decimal is 0.000027900...

Bring down a 0: 31000. Divide by 11111: 0. Write 0. Decimal is 0.0000279000...

Bring down a 0: 310000. Divide by 11111: 27. Write 27. Decimal is 0.0000279000027...

So, the decimal is 0.000027900002790000279... which is 0.0000279 repeating every 7 digits. But according to the fraction, it should be 0.0027900279... which is different. So, why is this happening?

Wait, let me check the value of 0.00002790000279... and 0.0027900279... Let me multiply 0.00002790000279... by 100000. Let me see:

0.00002790000279... * 100000 = 2.790000279...

Which is approximately 2.790000279...

On the other hand, 0.0027900279... * 100000 = 279.00279...

So, these are different. Therefore, one of the decimal expansions is incorrect.

But since we know that 31/11111 = 279/99999, and 279/99999 = 0.0027900279..., then the long division must have an error. Where is the error?

Wait, let me check the initial steps of the long division. When I divided 31 by 11111, I started with 31.000000... and brought down zeros. But perhaps I made a mistake in the number of zeros brought down?

Wait, let me try a different approach. Let me use the fact that 1/11111 = 0.\overline{00009}, so 31/11111 = 31 * 0.\overline{00009} = 0.\overline{00009 * 31} = 0.\overline{279}?

Wait, but 00009 * 31 = 279, so 0.\overline{279} would be 0.279279279..., but this is not matching with the previous results. However, if I consider that 0.\overline{00009} is 0.0000900009..., then multiplying by 31 gives 0.00009 * 31 = 0.00279, and then the rest of the decimal is 0.00000000009 * 31 = 0.000000000279, and so on. So, the decimal is 0.0027900002790000279..., which is 0.002790000279... which is 0.00279 repeating every 5 digits? Wait, but how?

Wait, if I write 0.0027900002790000279..., this is 0.002790000279..., which is 0.00279 followed by 0000279... So, the repeating part is "00279", which is 5 digits. But how does this fit with the long division?

Wait, let me check the value of 0.002790000279... as a fraction. Let me denote y = 0.002790000279...

Multiply y by 10^5 (since the repeating part is 5 digits): 10^5 * y = 279.00002790000279...

But the original y is 0.002790000279..., so 10^5 * y = 279 + y

Therefore, 10^5 * y - y = 279

y(10^5 - 1) = 279

y = 279 / (10^5 - 1) = 279 / 99999

Which is correct, since 279/99999 = 31/11111. So, this suggests that the correct decimal expansion is 0.002790000279..., which is 0.00279 repeating every 5 digits. But why does the long division give a different result?

Ah! I think I see the mistake. In the long division, I was dividing 31 by 11111, but I think I made a mistake in the number of zeros brought down. Let me try again.

Let me perform the long division of 31 divided by 11111, carefully.

Start with 31.000000...

11111 goes into 31 zero times. Write 0.

Bring down a 0: 310. Still 0. Write 0.

Bring down a 0: 3100. Still 0. Write 0.

Bring down a 0: 31000. Still 0. Write 0.

Bring down a 0: 310000. Now, divide 310000 by 11111.

As before, 11111 * 27 = 299997. Subtract: 310000 - 299997 = 10003. Write 27. So, the decimal is 0.000027...

Now, bring down a 0: 100030. Divide by 11111: 9. Write 9. Decimal is 0.0000279...

Now, bring down a 0: 31. Divide by 11111: 0. Write 0. Decimal is 0.00002790...

Now, bring down a 0: 310. Divide by 11111: 0. Write 0. Decimal is 0.000027900...

Now, bring down a 0: 3100. Divide by 11111: 0. Write 0. Decimal is 0.0000279000...

Now, bring down a 0: 31000. Divide by 11111: 0. Write 0. Decimal is 0.00002790000...

Now, bring down a 0: 310000. Divide by 11111: 27. Write 27. Decimal is 0.0000279000027...

Wait, but this suggests that the decimal is 0.00002790000279..., which is 0.0000279 repeating every 7 digits. But according to the fraction, it should be 0.002790000279... which is 0.00279 repeating every 5 digits. So, there is a contradiction.

Wait, let me check the value of 0.00002790000279... and 0.002790000279... Let me multiply 0.00002790000279... by 100000.

0.00002790000279... * 100000 = 2.790000279...

On the other hand, 0.002790000279... * 100000 = 279.0000279...

So, these are different. Therefore, one of the decimal expansions is incorrect. But since we know that 31/11111 = 279/99999 = 0.002790000279..., the long division must have an error.

Wait, let me check the long division again. When I brought down the first zero after 310000, I got 100030, divided by 11111 to get 9, remainder 31. Then, bringing down a zero gives 310, which is the same as the original numerator. Wait, but the original numerator was 31, and now it's 310. So, this suggests that the decimal repeats every 6 digits? Let me count the digits:

After the decimal, the digits are 0, 0, 0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9, ... So, the repeating part is "0000279", which is 7 digits. But this contradicts the fraction.

Wait, let me try to compute 0.00002790000279... as a fraction. Let me denote x = 0.00002790000279...

Let me multiply x by 10^7 to shift the decimal 7 places: 10^7 * x = 279.00002790000279...

But x = 0.00002790000279..., so 10^7 * x = 279 + x

Therefore, 10^7 * x - x = 279

x(10^7 - 1) = 279

x = 279 / (10^7 - 1) = 279 / 9999999

But we know that x = 31/11111 = 279/99999. So, 279/99999 = 279/9999999? No, this is not true. 99999 ≠ 9999999. So, this suggests that my long division is incorrect.

Wait, but how? Where is the mistake? Let me check the initial steps again.

When I divided 31 by 11111, I started with 31.000000... and brought down zeros. But perhaps I should have brought down more zeros or considered the decimal places differently. Let me try a different approach.

Let me note that 1/11111 = 0.\overline{00009}, so 31/11111 = 31 * 0.\overline{00009} = 0.\overline{00009 * 31} = 0.\overline{279}?

Wait, but 00009 * 31 = 279, so 0.\overline{279} is 0.279279279..., but this is not matching with the previous results. However, if I consider that 0.\overline{00009} is 0.0000900009..., then multiplying by 31 gives 0.00009 * 31 = 0.00279, and then the rest of the decimal is 0.00000000009 * 31 = 0.000000000279, and so on. So, the decimal is 0.0027900002790000279..., which is 0.00279 repeating every 5 digits.

But how does this relate to the long division? It seems like there is a discrepancy. Let me try to perform the long division again, but this time, I'll track the decimal places more carefully.

Let me write 31 divided by 11111 as 0.00002790000279... and see if it matches with the fraction.

Let me compute 0.00002790000279... * 11111. If this equals 31, then it's correct.

Let me compute 0.00002790000279... * 11111.

First, 0.00002790000279... * 11111 = 0.00002790000279... * 10000 + 0.00002790000279... * 1111

Wait, this might be complex. Let me approximate.

Let me take the decimal as 0.00002790000279... ≈ 0.00002790000279

Multiply by 11111:

0.00002790000279 * 11111 ≈ 0.00002790000279 * 10000 + 0.00002790000279 * 1111

= 0.2790000279 + 0.00002790000279 * 1111

≈ 0.2790000279 + 0.0310000279

≈ 0.3100000558

Which is approximately 0.31, which is 31/100, but we need it to be 31/11111 ≈ 0.00279. Wait, this suggests that my approximation is wrong. Wait, no, I think I made a mistake in the multiplication.

Wait, let me do it properly. Let me take x = 0.00002790000279...

Multiply x by 11111:

x * 11111 = 0.00002790000279... * 11111

Let me note that 0.00002790000279... = 279/9999999 (from earlier). So, 279/9999999 * 11111 = 279 * 11111 / 9999999.

Let me compute 279 * 11111 = 279 * 11111. Let me calculate this:

First, 279 * 10000 = 2,790,000

279 * 1000 = 279,000

279 * 100 = 27,900

279 * 10 = 2,790

279 * 1 = 279

Adding these together: 2,790,000 + 279,000 = 3,069,000; +27,900 = 3,096,900; +2,790 = 3,099,690; +279 = 3,099,969.

So, 279 * 11111 = 3,099,969.

Now, divide by 9999999: 3,099,969 / 9999999.

Simplify this fraction: Divide numerator and denominator by 3: 1,033,323 / 3,333,333. Divide by 3 again: 344,441 / 1,111,111. This is approximately 0.31, which is 31/100. But we expected it to be 31/11111. Wait, this suggests that 279/9999999 * 11111 = 31/100, which is not correct. This indicates a mistake in the earlier assumption.

Wait, this is getting too convoluted. Let me go back to the original fraction: 31/11111. Let me use a calculator approximation. Let me approximate 1/11111 ≈ 0.0000900009. So, 31 * 0.0000900009 ≈ 0.0027900279. So, the decimal is approximately 0.0027900279..., which is 0.00279 repeating every 5 digits. This suggests that the correct decimal expansion is 0.0027900279..., with a repeating cycle of 5 digits.

But why does the long division give a different result? I must have made a mistake in the long division steps. Let me try one more time, carefully.

Let me perform the long division of 31 divided by 11111, step by step, and track the decimal places.

1. 31 ÷ 11111 = 0.0...
2. 31.0 ÷ 11111 = 0.00...
3. 31.00 ÷ 11111 = 0.000...
4. 31.000 ÷ 11111 = 0.0000...
5. 31.0000 ÷ 11111 = 0.00000...
6. 31.00000 ÷ 11111: Now, 310000 ÷ 11111 = 27, remainder 10003. So, the first non-zero digit is 27, so the decimal is 0.000027...
7. Bring down a 0: 100030 ÷ 11111 = 9, remainder 31. So, next digit is 9, decimal is 0.0000279...
8. Bring down a 0: 310 ÷ 11111 = 0, remainder 310. Next digit 0, decimal is 0.00002790...
9. Bring down a 0: 3100 ÷ 11111 = 0, remainder 3100. Next digit 0, decimal is 0.000027900...
10. Bring down a 0: 31000 ÷ 11111 = 0, remainder 31000. Next digit 0, decimal is 0.0000279000...
11. Bring down a 0: 310000 ÷ 11111 = 27, remainder 10003. Next digit 27, decimal is 0.0000279000027...

Wait, so the decimal is 0.000027900002790000279... which is 0.0000279 repeating every 7 digits. But this contradicts the fraction.

However, if I take this decimal and multiply by 11111, I should get 31. Let me try:

Let x = 0.00002790000279...

Multiply x by 11111: 0.00002790000279... * 11111 = 0.00002790000279... * 10000 + 0.00002790000279... * 1111

= 0.2790000279... + 0.00002790000279... * 1111

Let me approximate 0.00002790000279... * 1111 ≈ 0.0000279 * 1111 ≈ 0.0310000000...

So, total is approximately 0.279 + 0.031 = 0.31, which is 31/100, not 31/11111. This suggests that the long division is incorrect, and the correct decimal is 0.0027900279..., which is 0.00279 repeating every 5 digits.

But why is there this discrepancy? It must be due to an error in the long division steps. Let me try to find where.

Wait, when I divided 31 by 11111, I started with 31.000000... and brought down zeros. But perhaps I should have considered that after the decimal, the first few digits are 0.0000279... but this is not matching with the fraction.

Alternatively, let me use the fact that 1/11111 = 0.\overline{00009}, so 31/11111 = 31 * 0.\overline{00009} = 0.\overline{00009 * 31} = 0.\overline{279}?

Wait, but 00009 * 31 = 279, so 0.\overline{279} is 0.279279279..., but this is not matching with the fraction. However, if I consider that 0.\overline{00009} is 0.0000900009..., then multiplying by 31 gives 0.00009 * 31 = 0.00279, and the rest of the decimal is 0.00000000009 * 31 = 0.000000000279, and so on. So, the decimal is 0.0027900002790000279..., which is 0.00279 repeating every 5 digits.

This suggests that the correct decimal expansion is 0.002790000279..., with a repeating cycle of 5 digits: "00279". But how does this fit with the long division? It seems like there is a miscalculation in the long division steps, possibly due to an error in the number of zeros brought down or in the division steps.

Given the time I've spent and the conflicting results, I think the correct approach is to use the fraction 279/99999 = 31/11111, and since 99999 = 10^5 - 1, the decimal expansion of 279/99999 is 0.\overline{00279}, which is 5 digits. Therefore, the repeating decimal has 5 digits.

But earlier, the long division suggested 7 digits, which must be incorrect. Therefore, I must have made a mistake in the long division steps. Given that the fraction approach is more reliable, I will go with the 5-digit repeating cycle.

Wait, but let me check with another example. Let me take 1/11111 = 0.\overline{00009}, which is 5 digits. Then, 2/11111 = 0.\overline{00018}, which is 5 digits. Similarly, 3/11111 = 0.\overline{00027}, and so on. So, 31/11111 should be 0.\overline{000279} if 31 * 00009 = 000279. Wait, 31 * 00009 = 000279, which is 279. So, 0.\overline{000279} is 0.000279000279..., which is 6 digits? Wait, no, 000279 is 6 digits. But 1/11111 is 0.\overline{00009} (5 digits). So, multiplying by 31 gives 0.\overline{000279} (6 digits)? But 000279 is 6 digits, so the repeating cycle is 6 digits. But this contradicts the earlier fraction approach.

Wait, this is getting too confusing. Let me use the fact that 31/11111 = 279/99999, and 99999 = 10^5 - 1. The decimal expansion of a fraction a/(10^n - 1) has a repeating cycle of n digits. So, 279/99999 has a repeating cycle of 5 digits. Therefore, the decimal expansion is 0.\overline{00279}, which is 5 digits.

But how does this match with the long division? It must be that in the long division, I made a mistake in the number of zeros or in the division steps. Given the time I've spent and the consistency of the fraction approach, I will conclude that the repeating decimal has 5 digits.

But wait, let me check with a smaller example. Let's take 1/11111 = 0.\overline{00009} (5 digits). Let's multiply by 10: 10/11111 = 0.\overline{00090} (5 digits). Similarly, 100/11111 = 0.\overline{00900}, and so on. So, multiplying by 31, which is 31 * 0.\overline{00009} = 0.\overline{00009 * 31} = 0.\overline{279}?

Wait, 00009 * 31 = 279, so it should be 0.\overline{279}, but this is 3 digits. But 0.\overline{279} is 0.279279279..., which is different from 0.0027900279...

Wait, this suggests that multiplying the repeating decimal by 31 does not simply multiply the repeating block, but instead shifts the decimal. This is because 0.\overline{00009} is 0.0000900009..., and multiplying by 31 gives 0.002790000279..., which is 0.002790000279..., which is 0.\overline{00279} (5 digits).

So, the repeating block is "00279", which is 5 digits. Therefore, the answer should be 5.

But why does the long division suggest 7 digits? I must have made a mistake in the long division steps. Given the time I've spent and the consistency of the fraction approach, I will conclude that the repeating decimal has 5 digits.

**Final Answer**
The number of repeating digits is \boxed{5}.

To determine how many digits repeat in the decimal expansion of the fraction $\frac{31}{11111}$, we begin by analyzing the structure of the denominator.

---

### Step 1: Understand the Denominator

The denominator is $11111$. This number is special because it can be expressed as:

$$
11111 = \frac{10^5 - 1}{9}
$$

This is because:

$$
\frac{10^5 - 1}{9} = \frac{99999}{9} = 11111
$$

So, we can rewrite the original fraction as:

$$
\frac{31}{11111} = \frac{31 \cdot 9}{99999} = \frac{279}{99999}
$$

---

### Step 2: Analyze the Decimal Expansion

Now, consider the fraction $\frac{279}{99999}$. Since the denominator is of the form $10^n - 1$, the decimal expansion of such a fraction is **repeating** and the **length of the repeating block** is equal to the number of digits in the denominator, which is 5.

So, the decimal expansion of $\frac{279}{99999}$ is a repeating decimal with a **5-digit repeating block**.

To confirm, we can write:

$$
\frac{279}{99999} = 0.\overline{00279}
$$

This means the repeating part is **"00279"**, which is **5 digits** long.

---

### Step 3: Final Conclusion

Since $\frac{31}{11111} = \frac{279}{99999}$, and the decimal expansion of $\frac{279}{99999}$ is a repeating decimal with a 5-digit repeating block, the number of digits that repeat in the decimal expansion of $\frac{31}{11111}$ is:

$$
\boxed{5}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many digits repeat in the decimal expansion of 31 divided by 11111. Let me start by recalling that when you divide a number by another, if the denominator has prime factors other than 2 and 5, the decimal will repeat. Since 11111 is the denominator here, I should check its prime factors to understand the repeating pattern.

First, let me try dividing 31 by 11111 directly. But maybe there's a smarter way than long division. Let me think. The denominator is 11111. Hmm, 11111... that number looks familiar. Wait, 11111 is 10^5 - 1 over 9? Let me check: 10^5 is 100000, so 100000 - 1 is 99999. Divided by 9 is 11111. Yes! So 11111 = (10^5 - 1)/9. Therefore, 1/11111 is 9/(10^5 - 1). So, 1/11111 is equal to 9/99999, which is 0.000090000900009..., repeating every 5 digits? Wait, but 9/99999 is 0.\overline{00009}? Let me verify that.

If I take 1/99999, that's 0.\overline{00001}, right? Because 1/99999 = 0.000010000100001..., repeating every 5 digits. Then multiplying by 9 would give 0.000090000900009..., so the repeating part is "00009", which is 5 digits. Therefore, 1/11111 is 0.\overline{00009}, so the repeating decimal has 5 digits. Therefore, 31/11111 would be 31 times that decimal. So, 31 * 0.\overline{00009}.

But how does multiplying by 31 affect the repeating decimal? Let me think. If I have 0.\overline{00009}, multiplying by 31 would give me a decimal that's 31 times that. Let me try to compute that. Alternatively, maybe there's a pattern here. Let me think about how multiplying by 31 would affect the repeating sequence.

Alternatively, maybe I can perform the division directly. Let me try dividing 31 by 11111. Since 31 is less than 11111, the decimal will start with 0. Let me write this out as a long division problem.

So, 31 divided by 11111. Let me add decimal places and zeros to 31. So, 31.000000...

First, 11111 goes into 310 zero times. Then 3100, still zero. 31000, still zero. 310000 divided by 11111. Let me calculate how many times 11111 fits into 310000.

Let me compute 11111 * 2 = 22222. 11111 * 3 = 33333. But 33333 is larger than 310000? Wait, no. Wait, 11111 * 2 = 22222. 11111 * 27 = ?

Wait, maybe I need to compute 310000 divided by 11111. Let me approximate. 11111 * 2 = 22222. 11111 * 28 = 11111 * 27 + 11111. Let me compute 11111 * 27. 11111 * 20 = 222220, 11111 * 7 = 77777. Adding them together: 222220 + 77777 = 299,997. Then adding another 11111 gives 299,997 + 11,111 = 311,108. But 311,108 is larger than 310,000. So 11111 * 27 = 299,997. So 299,997 is less than 310,000. The difference is 310,000 - 299,997 = 10,003. So, 11111 goes into 310,000 27 times with a remainder of 10,003. Therefore, the first digit after the decimal is 0. Then, after the decimal, we have 0. So, first, we have 0. Then, after the decimal, we have 0. Then, we bring down zeros. Wait, maybe I need to track this more carefully.

Wait, let me start over. Dividing 31 by 11111:

31 ÷ 11111 = 0. So, we write 0. and then proceed with the decimal.

Multiply 31 by 10: 310. Still less than 11111. Next digit 0.

Multiply by 10 again: 3100. Still less. Next digit 0.

Multiply by 10 again: 31000. Still less. Next digit 0.

Multiply by 10 again: 310000. Now, divide 310000 by 11111. As above, 11111 * 27 = 299,997. So, 27. Then subtract 299,997 from 310,000: 310,000 - 299,997 = 10,003. Bring down a zero: 100,030.

Now divide 100,030 by 11111. Let me see how many times 11111 fits into 100,030. Let me compute 11111 * 9 = 99,999. That's very close. So 9 times. Subtract 99,999 from 100,030: 100,030 - 99,999 = 31. Bring down a zero: 310.

Now, we have 310 again. Wait, this is the same as the original numerator. So, the decimal repeats here. So, the decimal expansion is 0.000279... and then repeats?

Wait, let me track the digits. Let me write down the steps:

First, after the decimal, we had four zeros (from multiplying by 10 four times), then we had 27, then 9, then we get back to 310, which is where we started. So, the decimal expansion is 0.000279000279000279..., repeating every 6 digits? Wait, let me check.

Wait, let me retrace:

Starting with 31.000000...

First division steps:

- 31 divided by 11111: 0.
- 310 divided by 11111: 0.
- 3100 divided by 11111: 0.
- 31000 divided by 11111: 0.
- 310000 divided by 11111: 27 (as 27*11111=299997), remainder 10003.
- Bring down a 0: 100030 divided by 11111: 9 (9*11111=99999), remainder 31.
- Bring down a 0: 310 divided by 11111: 0.
- Bring down a 0: 3100 divided by 11111: 0.
- Bring down a 0: 31000 divided by 11111: 0.
- Bring down a 0: 310000 divided by 11111: 27 again...

Wait, so after the first 27 and 9, we get back to 310, which is the same as the original numerator. Therefore, the decimal expansion is 0.000279000279000279..., so the repeating part is "000279"? Wait, but let me check the digits.

Wait, let me write down the decimal digits step by step:

After the decimal point, the first four digits are 0s (from the first four divisions). Then, when we divided 310000 by 11111, we got 27, so the fifth digit is 2, sixth is 7. Then, after subtracting, we had remainder 10003, then bringing down a zero gives 100030. Dividing that by 11111 gives 9, so the seventh digit is 9. Then remainder 31, bringing down a zero gives 310. Then, we have to divide 310 by 11111, which is 0, so the eighth digit is 0. Then bring down a zero to get 3100, which is 0, ninth digit 0. Bring down a zero to get 31000, which is 0, tenth digit 0. Bring down a zero to get 310000 again, which is 27, so the eleventh digit is 2, twelfth is 7, and so on.

So the decimal expansion is 0.000279000279000279..., so the repeating part is "000279", which is 6 digits. Wait, but let me check if that's correct. Let me count the digits between the repeating parts. From the first 0.000279... then the next digits are 000279... So, the repeating cycle is 000279, which is 6 digits. Therefore, the answer would be 6?

But earlier, I thought that 1/11111 is 0.\overline{00009}, which is 5 digits. But when multiplying by 31, the decimal becomes 0.\overline{000279}? Wait, but 0.\overline{000279} would be 000279 repeating. Let me check if that's correct.

Alternatively, maybe there's a mistake here. Let me verify with another approach. Since 1/11111 = 0.\overline{00009}, then multiplying by 31 would give 31 * 0.\overline{00009} = 0.\overline{00009 * 31}?

Wait, but multiplying a repeating decimal by a number doesn't necessarily just multiply the repeating part. For example, 0.\overline{abc} * n is not necessarily 0.\overline{abc * n} unless n is a single digit. So maybe my previous approach is flawed.

Alternatively, let me think of 31/11111 as 31/(10^5 - 1)/9. Wait, earlier I said 11111 = (10^5 - 1)/9. Therefore, 31/11111 = 31 * 9 / (10^5 - 1) = 279 / 99999. Therefore, 279 divided by 99999. Which is 0.\overline{00279}? Wait, 279/99999. Let me see. Since 279/99999 = 0.002790027900279..., so the repeating part is "00279", which is 5 digits? Wait, but 279 divided by 99999. Let me check:

If I have 279 divided by 99999, then since 99999 is 10^5 - 1, so 279/(10^5 - 1) = 0.\overline{00279}. Wait, but how many digits? Let me think. If I have a fraction a/(10^n - 1), then the decimal repeats every n digits. So here, 279/(10^5 - 1) would have a repeating cycle of 5 digits. But 279 is a 3-digit number. So, how does that work?

Wait, for example, 1/99999 = 0.\overline{00001}, which is 5 digits. Similarly, 279/99999 would be 0.\overline{00279}, which is 5 digits. But 00279 is 5 digits. So, the decimal expansion is 0.002790027900279..., so the repeating part is 00279, which is 5 digits. But earlier, when I did the long division, I thought it was 0.000279000279..., which is 6 digits. There's a contradiction here. Which one is correct?

Wait, let me check with actual calculation. Let me compute 279 divided by 99999. Let me do that division. 279 ÷ 99999. Since 279 is less than 99999, it's 0. Then, 279.000000...

Multiply by 10: 2790. Still less than 99999. Next digit 0.

Multiply by 10: 27900. Still less. Next digit 0.

Multiply by 10: 279000. Still less. Next digit 0.

Multiply by 10: 2790000. Now divide by 99999. Let me see how many times 99999 fits into 2790000.

Well, 99999 * 27 = 2699973. Let me check 99999 * 27:

99999 * 20 = 1,999,980

99999 * 7 = 699,993

Adding them: 1,999,980 + 699,993 = 2,699,973. Then, 2,699,973. Subtract that from 2,790,000: 2,790,000 - 2,699,973 = 90,027. Bring down a zero: 900,270.

Divide 900,270 by 99,999. Let me see, 99,999 * 9 = 899,991. Subtract that from 900,270: 900,270 - 899,991 = 279. Bring down a zero: 2790. Now we are back to where we started. So the decimal expansion is 0.000279000279..., so the repeating part is "000279", which is 6 digits. Wait, but according to the previous logic, it should be 5 digits. What's going on here?

Wait, maybe my assumption that 279/99999 is 0.\overline{00279} is wrong. Let me check with another approach. Let me note that 279/99999 = (279/9)/ (11111) = 31/11111. Which is the original fraction. So, if I have 279/99999, which is equal to 0.000279000279..., then the repeating cycle is 6 digits. But why does the formula a/(10^n - 1) have a repeating cycle of n digits? Maybe because if the numerator has fewer digits, it's padded with leading zeros. For example, 279/99999 is equivalent to 00279/99999, so the decimal is 0.0027900279..., but that would be 5 digits. Wait, but in reality, when I did the division, I got 0.000279000279..., which is 6 digits. There's a contradiction here. Which is correct?

Wait, let me check with a calculator. Let me compute 31 divided by 11111. Let me approximate. 11111 * 0.00279 = ?

Well, 11111 * 0.002 = 22.222

11111 * 0.00079 = ?

First, 11111 * 0.0007 = 7.7777

11111 * 0.00009 = 0.99999

So total is 7.7777 + 0.99999 ≈ 8.77769

So total 0.00279 * 11111 ≈ 22.222 + 8.77769 ≈ 30.99969, which is approximately 31. So that suggests that 31/11111 ≈ 0.0027900279..., so the decimal is 0.0027900279..., which would be repeating every 5 digits? Wait, but according to the long division, it was 0.000279000279... which is different. Wait, there's confusion here. Let me check again.

Wait, if 0.0027900279... is the decimal, then that would be 0.00279 repeating, which is 5 digits. But according to the long division, I had 0.000279000279..., which is 0.000279 repeating, which is 6 digits. Which is correct?

Wait, let me check with another method. Let me compute 31/11111 using another approach. Let me note that 1/11111 = 0.\overline{00009} as I thought earlier. Then 31 * 0.\overline{00009} = ?

Multiplying a repeating decimal by an integer. Let me think. If I have 0.\overline{00009} = 0.000090000900009..., then multiplying by 31:

Let me write it as 0.000090000900009... * 31.

Let me compute 0.00009 * 31 = 0.00009 * 30 + 0.00009 * 1 = 0.0027 + 0.00009 = 0.00279.

Then, the rest of the decimal is 0.0000000000900009... * 31, which would be 0.000000000279..., so adding that to 0.00279 gives 0.002790000279..., so the decimal is 0.002790000279..., which is 0.00279 repeating? Wait, but the decimal is 0.002790000279..., so the repeating part is "0000279"? Wait, no. Wait, if you have 0.002790000279..., then the repeating part is "0000279"? Wait, but how?

Wait, maybe I need to think of the decimal expansion as follows. If 1/11111 = 0.\overline{00009}, then multiplying by 31 gives 0.\overline{00009 * 31}?

But 00009 * 31 is 279. So, 0.\overline{279}? But that would be 0.279279279..., but that's not matching with the previous calculation. Wait, there's confusion here. Let me think again.

If I have 0.\overline{abc}, then multiplying by n gives a decimal that depends on the value of n. For example, multiplying by 31 would not just multiply the repeating block. However, if the repeating block is of length k, then multiplying by n could result in a different repeating block. However, in this case, since 1/11111 = 0.\overline{00009}, which is 5 digits, multiplying by 31 would give a decimal that is 31 * 0.\overline{00009} = 0.\overline{00009 * 31} only if 00009 * 31 doesn't cause a carryover that affects the repeating block. But 00009 * 31 is 279, which is 3 digits. So, if we have 0.\overline{00009} * 31 = 0.\overline{279}?

But that would be 0.279279279..., but that contradicts the previous calculation where we had 0.00279... So, clearly, there's a mistake here. Where is the error?

Ah! Wait, 0.\overline{00009} is not 0.0000900009..., but rather 0.0000900009... which is 0.00009 repeating. So, multiplying by 31 would be 0.00009 * 31 = 0.00279, and then the rest of the decimal would be 0.00000000009 * 31 = 0.000000000279, etc. So, the decimal expansion is 0.0027900002790000279..., which is 0.002790000279..., so the repeating part is "0000279"? Wait, but how?

Alternatively, maybe the decimal expansion is 0.002790000279..., which is 0.00279 repeating? But that would require that the decimal is 0.002790000279..., which is not a simple repeating of 00279. Wait, maybe the repeating cycle is longer. Let me check with the long division result.

Earlier, when I did the long division, I found that the decimal expansion is 0.000279000279..., which is 0.000279 repeating. But according to the multiplication approach, it's 0.00279... So which is correct?

Wait, let me check with actual decimal multiplication. Let me compute 31/11111 numerically. Let me approximate 1/11111 ≈ 0.00009. Then 31 * 0.00009 = 0.00279. But if I compute 31/11111, let me do a better approximation. Let me compute 1/11111 ≈ 0.0000900009... So multiplying by 31 gives 0.0000900009... * 31 = 0.0027900279..., which is 0.0027900279..., so the decimal is 0.0027900279..., which is 0.00279 repeating? Wait, but the decimal is 0.0027900279..., so the repeating part is "00279"? Wait, but how?

Wait, if I write 0.002790027900279..., then the repeating part is "00279", which is 5 digits. But in the long division, I had 0.000279000279..., which is 0.000279 repeating, which is 6 digits. There's a contradiction here. Which one is correct?

Wait, let me check with another method. Let me compute 31 divided by 11111 using a calculator. Since I don't have a calculator here, let me try to compute it more carefully.

Let me consider that 1/11111 ≈ 0.000090000900009...

So multiplying by 31:

First, 0.00009 * 31 = 0.00279

Then, the next part is 0.00000000009 * 31 = 0.000000000279

Adding these together: 0.00279 + 0.000000000279 = 0.002790000279...

So, the decimal expansion is 0.002790000279..., which is 0.002790000279..., so the repeating part is "0000279"? Wait, but how?

Wait, if I write it out:

0.0027900002790000279...

So, after the decimal, the digits are:

0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9,...

So, the repeating part is "0000279", which is 7 digits? Wait, but that seems inconsistent. Wait, let me count:

Starting from the decimal point:

Position 1: 0

Position 2: 0

Position 3: 2

Position 4: 7

Position 5: 9

Position 6: 0

Position 7: 0

Position 8: 0

Position 9: 0

Position 10: 2

Position 11: 7

Position 12: 9

Position 13: 0

Position 14: 0

Position 15: 0

Position 16: 0

Position 17: 2

...

So, the pattern is "00279" followed by four zeros? Wait, no. Wait, from position 1-5: 0,0,2,7,9; then positions 6-9: 0,0,0,0; then positions 10-14: 2,7,9,0,0; then positions 15-19: 0,0,0,0,2... Wait, this seems inconsistent. Maybe my previous assumption is wrong.

Alternatively, maybe there's a mistake in the long division. Let me go back to the long division steps.

Earlier, I had:

After dividing 31 by 11111, I had:

- 31.000000...

First four divisions gave 0.0000...

Then, dividing 310000 by 11111 gave 27, remainder 10003.

Then, bringing down a zero: 100030 divided by 11111 is 9, remainder 31.

Then, bringing down a zero: 310 divided by 11111 is 0, remainder 310.

Then, bringing down a zero: 3100 divided by 11111 is 0, remainder 3100.

Then, bringing down a zero: 31000 divided by 11111 is 0, remainder 31000.

Then, bringing down a zero: 310000 divided by 11111 is 27, remainder 10003...

So, the decimal digits obtained are:

After the decimal: 0, 0, 0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9,...

So, the decimal expansion is 0.000027900002790000279..., so the repeating part is "0000279", which is 7 digits? Wait, but how?

Wait, let me count the digits between the repeating parts. From the first 0.0000279... then the next digits are 0000279... So, the repeating cycle is "0000279", which is 7 digits? But that seems inconsistent with the previous thought that it should be 5 digits. What's going on here?

Alternatively, maybe there's a mistake in the long division steps. Let me check again:

Starting with 31 divided by 11111:

- 31.000000...

First, 11111 goes into 31 zero times. Write 0.

Multiply by 10: 310. Still 0. Write 0.

Multiply by 10: 3100. Still 0. Write 0.

Multiply by 10: 31000. Still 0. Write 0.

Multiply by 10: 310000. Now divide by 11111.

As before, 11111 * 27 = 299,997. Subtract from 310,000: 310,000 - 299,997 = 10,003. Write down 27. So, the decimal so far is 0.000027...

Bring down a 0: 100,030. Divide by 11111. 11111 * 9 = 99,999. Subtract: 100,030 - 99,999 = 31. Write down 9. Decimal is 0.0000279...

Bring down a 0: 310. Divide by 11111: 0. Write down 0. Decimal is 0.00002790...

Bring down a 0: 3100. Divide by 11111: 0. Write down 0. Decimal is 0.000027900...

Bring down a 0: 31000. Divide by 11111: 0. Write down 0. Decimal is 0.0000279000...

Bring down a 0: 310000. Divide by 11111: 27. Write down 27. Decimal is 0.0000279000027...

So, the repeating cycle is "0000279", which is 7 digits? Wait, but the digits after the decimal are:

0.000027900002790000279...

So, the repeating part is "0000279", which is 7 digits? Wait, but how? Because after the first 0.0000279, we have 0000279 again. Wait, but the first "0000279" is positions 1-7: 0,0,0,0,2,7,9. Then the next digits are 0,0,0,0,2,7,9... So, the repeating cycle is 7 digits? But that seems odd. However, according to the long division, that's what it is.

But earlier, when I thought of 279/99999, I thought it would be 0.\overline{00279}, which is 5 digits. But according to the long division, it's 0.\overline{0000279}, which is 7 digits. There's a contradiction here. Where is the mistake?

Wait, let me check 279 divided by 99999. If I write 279/99999, that's equal to 0.002790027900279..., which is 0.00279 repeating every 5 digits? Wait, but how?

Wait, 279/99999 = 0.002790027900279... So, the decimal is 0.0027900279..., which is 0.00279 repeating? But the repeating part is "00279", which is 5 digits. However, in the long division, I have 0.00002790000279..., which is different. So, which one is correct?

Wait, there must be a miscalculation in one of the methods. Let me check with fractions. Let me compute 279/99999. Simplify the fraction. Let's see if 279 and 99999 have common factors.

Divide numerator and denominator by 3: 279 ÷ 3 = 93; 99999 ÷ 3 = 33333. So, 93/33333. Divide by 3 again: 93 ÷ 3 = 31; 33333 ÷ 3 = 11111. So, 31/11111, which is the original fraction. So, 279/99999 = 31/11111. Therefore, both methods are consistent. So, if 279/99999 = 0.0027900279..., then why does the long division give me 0.00002790000279...?

Wait, there must be an error in my long division steps. Let me check again.

Wait, when I did the long division earlier, I had:

After dividing 31 by 11111, I had:

- 31.000000...

First four divisions gave 0.0000...

Then, dividing 310000 by 11111 gave 27, remainder 10003. So, the decimal digits so far are 0.000027...

Then, bringing down a zero: 100030 divided by 11111 is 9, remainder 31. So, decimal is 0.0000279...

Then, bringing down a zero: 310 divided by 11111 is 0, remainder 310. So, decimal is 0.00002790...

Then, bringing down a zero: 3100 divided by 11111 is 0, remainder 3100. Decimal is 0.000027900...

Then, bringing down a zero: 31000 divided by 11111 is 0, remainder 31000. Decimal is 0.0000279000...

Then, bringing down a zero: 310000 divided by 11111 is 27, remainder 10003. Decimal is 0.0000279000027...

So, the decimal expansion is 0.000027900002790000279..., which is 0.0000279 repeating? Wait, but how?

Wait, if I write this out, it's 0.000027900002790000279..., so the repeating part is "0000279", which is 7 digits. But according to the fraction 279/99999, it should be 0.0027900279..., which is different. Where is the mistake?

Wait, maybe I made a mistake in the long division steps. Let me check the first few digits again.

Starting with 31 divided by 11111:

- 31 ÷ 11111 = 0. So, write 0.
- 31.0 ÷ 11111: 0.0
- 31.00 ÷ 11111: 0.00
- 31.000 ÷ 11111: 0.000
- 31.0000 ÷ 11111: 0.0000
- Now, 31.00000 ÷ 11111: Let's consider 310000 ÷ 11111.

Wait, here's the problem! When I did the long division, I considered 31.00000... as 31.00000, but when I multiply by 10^5, it's 3100000? Wait, no. Wait, let me clarify:

When performing long division, after the decimal point, each step is multiplying by 10. So starting with 31.000000...

First division step: 31 ÷ 11111 = 0, remainder 31. Bring down a 0: 310.

Second division step: 310 ÷ 11111 = 0, remainder 310. Bring down a 0: 3100.

Third division step: 3100 ÷ 11111 = 0, remainder 3100. Bring down a 0: 31000.

Fourth division step: 31000 ÷ 11111 = 0, remainder 31000. Bring down a 0: 310000.

Fifth division step: 310000 ÷ 11111 = 27, remainder 10003. So, the fifth digit after the decimal is 2, sixth is 7. So, decimal is 0.000027...

Then, bring down a 0: 100030 ÷ 11111 = 9, remainder 31. So, seventh digit is 9. Decimal is 0.0000279...

Bring down a 0: 310 ÷ 11111 = 0, remainder 310. Eighth digit is 0. Decimal is 0.00002790...

Bring down a 0: 3100 ÷ 11111 = 0, remainder 3100. Ninth digit is 0. Decimal is 0.000027900...

Bring down a 0: 31000 ÷ 11111 = 0, remainder 31000. Tenth digit is 0. Decimal is 0.0000279000...

Bring down a 0: 310000 ÷ 11111 = 27, remainder 10003. Eleventh digit is 2, twelfth is 7. Decimal is 0.0000279000027...

So, the decimal expansion is 0.000027900002790000279..., which is 0.0000279 repeating every 7 digits? Wait, but how does this relate to 279/99999?

Wait, 279/99999 = 0.002790027900279... So, why is there a discrepancy?

Wait, maybe I made a mistake in the long division. Let me check the value of 0.00002790000279... versus 0.0027900279...

Let me convert 0.00002790000279... to a fraction. Let me denote x = 0.00002790000279...

Multiply x by 10^7 (since the repeating cycle is 7 digits): 10^7 * x = 2790000279... Wait, no. Wait, if the repeating cycle is 7 digits, then multiplying by 10^7 would shift the decimal 7 places. Let me try:

Let me assume the repeating cycle is "0000279", which is 7 digits. So, x = 0.000027900002790000279...

Let me write x = 0.000027900002790000279...

Let me multiply x by 10^7 to shift the decimal 7 places:

10^7 * x = 279.00002790000279...

But the original x is 0.00002790000279...

So, 10^7 * x = 279 + x

Therefore, 10^7 * x = 279 + x

Subtract x: 10^7 * x - x = 279

x(10^7 - 1) = 279

x = 279 / (10^7 - 1) = 279 / 9999999

But earlier, we have x = 31/11111 = 279/99999. But 279/99999 is not equal to 279/9999999. So, this suggests that my assumption about the repeating cycle being 7 digits is wrong. Therefore, there must be an error in the long division steps.

Wait, this is confusing. Let me check with the fraction 279/99999. Let me compute 279 divided by 99999. Let me do this division properly.

Let me write 279 ÷ 99999. Since 279 is less than 99999, it's 0. So, 0. and then proceed.

Multiply by 10: 2790. Still less than 99999. Next digit 0.

Multiply by 10: 27900. Still less. Next digit 0.

Multiply by 10: 279000. Still less. Next digit 0.

Multiply by 10: 2790000. Now divide by 99999.

How many times does 99999 go into 2790000? Let me calculate 99999 * 27 = 2699973. Subtract from 2790000: 2790000 - 2699973 = 90027. Bring down a 0: 900270.

Divide 900270 by 99999. 99999 * 9 = 899991. Subtract: 900270 - 899991 = 279. Bring down a 0: 2790. Now we are back to where we started. So, the decimal expansion is 0.00002790000279..., which is 0.0000279 repeating every 6 digits? Wait, no. Wait, the digits after the decimal are:

0.000027900002790000279...

So, the repeating part is "0000279", which is 7 digits? But according to this division, after the first 0.0000279, we have 0000279 again. Wait, but the remainder after the first division was 279, and then we brought down a zero to get 2790, which is the same as the original numerator. Wait, no. Wait, the original numerator was 279, and after bringing down a zero, it became 2790, which is different. Wait, no, in this case, the original numerator was 279, and after multiplying by 10, it became 2790, then 27900, etc. But when we divided 2790000 by 99999, we got 27 with remainder 90027, then brought down a zero to get 900270, divided by 99999 to get 9 with remainder 279, then brought down a zero to get 2790, which is the same as the original numerator multiplied by 10. Wait, no, the original numerator was 279, and now it's 2790. So, this suggests that the decimal repeats every 6 digits? Wait, let me count the digits:

Starting from 0.0000279...

After the decimal, the digits are:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 0

Position 5: 2

Position 6: 7

Position 7: 9

Position 8: 0

Position 9: 0

Position 10: 0

Position 11: 0

Position 12: 2

Position 13: 7

Position 14: 9

...

So, the repeating cycle is "0000279", which is 7 digits. But according to the fraction, it should be 279/99999 = 0.0027900279..., which is different. Wait, but if I write 0.00002790000279..., that is equal to 0.00002790000279... which is 0.0000279 repeating every 7 digits. But how does this relate to 279/99999?

Wait, let me compute 0.00002790000279... as a fraction. Let me denote x = 0.00002790000279...

Let me multiply x by 10^7 to shift the decimal 7 places: 10^7 * x = 279.00002790000279...

But the original x is 0.00002790000279..., so 10^7 * x = 279 + x

Therefore, 10^7 * x - x = 279

x(10^7 - 1) = 279

x = 279 / (10^7 - 1) = 279 / 9999999

But earlier, we have x = 31/11111 = 279/99999. But 279/99999 is not equal to 279/9999999. So, this suggests that there is a contradiction, which means my long division must have an error.

Wait, but earlier, we saw that 31/11111 = 279/99999. So, if x = 279/99999, then x = 0.0027900279..., which is different from the long division result. So, where is the mistake?

Wait, let me check the value of 279/99999. Let me compute 279 divided by 99999. Let me do this division properly.

Let me write 279 ÷ 99999. Since 279 is less than 99999, it's 0. So, 0. and then proceed.

Multiply by 10: 2790. Still less than 99999. Next digit 0.

Multiply by 10: 27900. Still less. Next digit 0.

Multiply by 10: 279000. Still less. Next digit 0.

Multiply by 10: 2790000. Now divide by 99999.

As before, 99999 * 27 = 2699973. Subtract from 2790000: 2790000 - 2699973 = 90027. Bring down a 0: 900270.

Divide 900270 by 99999: 99999 * 9 = 899991. Subtract: 900270 - 899991 = 279. Bring down a 0: 2790. Now, this is the same as the original numerator multiplied by 10. Wait, original numerator was 279, now it's 2790. So, this suggests that the decimal repeats every 6 digits? Wait, let me count the digits:

After the decimal, the digits are:

0.00002790000279...

Wait, but how? Let me write out the decimal places:

Position 1: 0

Position 2: 0

Position 3: 0

Position 4: 0

Position 5: 2

Position 6: 7

Position 7: 9

Position 8: 0

Position 9: 0

Position 10: 0

Position 11: 0

Position 12: 2

Position 13: 7

Position 14: 9

...

So, the repeating cycle is "0000279", which is 7 digits. But according to the fraction, it should be 0.0027900279..., which is different. Wait, but if I write 0.00002790000279... as 0.00002790000279..., this is equal to 0.0000279 + 0.0000000000279 + ... which is a geometric series. Let me compute the sum.

Let me denote x = 0.00002790000279...

This can be written as x = 0.0000279 + 0.0000000000279 + 0.0000000000000000279 + ...

This is a geometric series with first term a = 0.0000279 and common ratio r = 10^{-6} (since each term is 10^{-6} times the previous one). So, the sum is a / (1 - r) = 0.0000279 / (1 - 10^{-6}) ≈ 0.0000279 / 0.999999 ≈ 0.00002790000279..., which is consistent with the long division. But how does this relate to 279/99999?

Wait, let me compute 279/99999. Let me do this division properly. Let me write 279 divided by 99999.

Let me note that 279/99999 = (279/9)/ (11111) = 31/11111, which is the original fraction. So, if I compute 31/11111, it should be equal to 279/99999, which is equal to 0.0027900279... or 0.00002790000279...? There's a contradiction here.

Wait, let me compute 279/99999 numerically. Let me approximate it. Let me note that 1/99999 ≈ 0.000010000100001... So, 279 * 0.000010000100001... = 0.002790027900279... So, this is 0.0027900279..., which is 0.00279 repeating every 5 digits. But according to the long division, it's 0.00002790000279..., which is different. So, there must be a mistake in the long division steps.

Wait, let me check the long division again. When I did the long division for 31/11111, I got 0.00002790000279..., but according to the fraction, it should be 0.0027900279... So, where is the error?

Wait, let me check the value of 0.00002790000279... and 0.0027900279... Let me convert both to fractions.

First, 0.00002790000279... = 279/9999999 (as per earlier calculation). Let me compute 279/9999999. Simplify this fraction. Divide numerator and denominator by 3: 93/3333333. Divide by 3 again: 31/1111111. So, 31/1111111. But the original fraction is 31/11111. So, these are different. Therefore, there must be a mistake in the long division.

Wait, this suggests that my long division was incorrect. Let me try again.

Let me perform the long division of 31 divided by 11111 carefully.

Start with 31.000000...

11111 goes into 31 zero times. Write 0.

Bring down a 0: 310. Still 0. Write 0.

Bring down a 0: 3100. Still 0. Write 0.

Bring down a 0: 31000. Still 0. Write 0.

Bring down a 0: 310000. Now, divide 310000 by 11111.

As before, 11111 * 27 = 299997. Subtract from 310000: 310000 - 299997 = 10003. Write 27. So, the decimal is 0.000027...

Bring down a 0: 100030. Divide by 11111. 11111 * 9 = 99999. Subtract: 100030 - 99999 = 31. Write 9. Decimal is 0.0000279...

Bring down a 0: 310. Divide by 11111: 0. Write 0. Decimal is 0.00002790...

Bring down a 0: 3100. Divide by 11111: 0. Write 0. Decimal is 0.000027900...

Bring down a 0: 31000. Divide by 11111: 0. Write 0. Decimal is 0.0000279000...

Bring down a 0: 310000. Divide by 11111: 27. Write 27. Decimal is 0.0000279000027...

So, the decimal is 0.000027900002790000279... which is 0.0000279 repeating every 7 digits. But according to the fraction, it should be 0.0027900279... which is different. So, why is this happening?

Wait, let me check the value of 0.00002790000279... and 0.0027900279... Let me multiply 0.00002790000279... by 100000. Let me see:

0.00002790000279... * 100000 = 2.790000279...

Which is approximately 2.790000279...

On the other hand, 0.0027900279... * 100000 = 279.00279...

So, these are different. Therefore, one of the decimal expansions is incorrect.

But since we know that 31/11111 = 279/99999, and 279/99999 = 0.0027900279..., then the long division must have an error. Where is the error?

Wait, let me check the initial steps of the long division. When I divided 31 by 11111, I started with 31.000000... and brought down zeros. But perhaps I made a mistake in the number of zeros brought down?

Wait, let me try a different approach. Let me use the fact that 1/11111 = 0.\overline{00009}, so 31/11111 = 31 * 0.\overline{00009} = 0.\overline{00009 * 31} = 0.\overline{279}?

Wait, but 00009 * 31 = 279, so 0.\overline{279} would be 0.279279279..., but this is not matching with the previous results. However, if I consider that 0.\overline{00009} is 0.0000900009..., then multiplying by 31 gives 0.00009 * 31 = 0.00279, and then the rest of the decimal is 0.00000000009 * 31 = 0.000000000279, and so on. So, the decimal is 0.0027900002790000279..., which is 0.002790000279... which is 0.00279 repeating every 5 digits? Wait, but how?

Wait, if I write 0.0027900002790000279..., this is 0.002790000279..., which is 0.00279 followed by 0000279... So, the repeating part is "00279", which is 5 digits. But how does this fit with the long division?

Wait, let me check the value of 0.002790000279... as a fraction. Let me denote y = 0.002790000279...

Multiply y by 10^5 (since the repeating part is 5 digits): 10^5 * y = 279.00002790000279...

But the original y is 0.002790000279..., so 10^5 * y = 279 + y

Therefore, 10^5 * y - y = 279

y(10^5 - 1) = 279

y = 279 / (10^5 - 1) = 279 / 99999

Which is correct, since 279/99999 = 31/11111. So, this suggests that the correct decimal expansion is 0.002790000279..., which is 0.00279 repeating every 5 digits. But why does the long division give a different result?

Ah! I think I see the mistake. In the long division, I was dividing 31 by 11111, but I think I made a mistake in the number of zeros brought down. Let me try again.

Let me perform the long division of 31 divided by 11111, carefully.

Start with 31.000000...

11111 goes into 31 zero times. Write 0.

Bring down a 0: 310. Still 0. Write 0.

Bring down a 0: 3100. Still 0. Write 0.

Bring down a 0: 31000. Still 0. Write 0.

Bring down a 0: 310000. Now, divide 310000 by 11111.

As before, 11111 * 27 = 299997. Subtract: 310000 - 299997 = 10003. Write 27. So, the decimal is 0.000027...

Now, bring down a 0: 100030. Divide by 11111: 9. Write 9. Decimal is 0.0000279...

Now, bring down a 0: 31. Divide by 11111: 0. Write 0. Decimal is 0.00002790...

Now, bring down a 0: 310. Divide by 11111: 0. Write 0. Decimal is 0.000027900...

Now, bring down a 0: 3100. Divide by 11111: 0. Write 0. Decimal is 0.0000279000...

Now, bring down a 0: 31000. Divide by 11111: 0. Write 0. Decimal is 0.00002790000...

Now, bring down a 0: 310000. Divide by 11111: 27. Write 27. Decimal is 0.0000279000027...

Wait, but this suggests that the decimal is 0.00002790000279..., which is 0.0000279 repeating every 7 digits. But according to the fraction, it should be 0.002790000279... which is 0.00279 repeating every 5 digits. So, there is a contradiction.

Wait, let me check the value of 0.00002790000279... and 0.002790000279... Let me multiply 0.00002790000279... by 100000.

0.00002790000279... * 100000 = 2.790000279...

On the other hand, 0.002790000279... * 100000 = 279.0000279...

So, these are different. Therefore, one of the decimal expansions is incorrect. But since we know that 31/11111 = 279/99999 = 0.002790000279..., the long division must have an error.

Wait, let me check the long division again. When I brought down the first zero after 310000, I got 100030, divided by 11111 to get 9, remainder 31. Then, bringing down a zero gives 310, which is the same as the original numerator. Wait, but the original numerator was 31, and now it's 310. So, this suggests that the decimal repeats every 6 digits? Let me count the digits:

After the decimal, the digits are 0, 0, 0, 0, 2, 7, 9, 0, 0, 0, 0, 2, 7, 9, ... So, the repeating part is "0000279", which is 7 digits. But this contradicts the fraction.

Wait, let me try to compute 0.00002790000279... as a fraction. Let me denote x = 0.00002790000279...

Let me multiply x by 10^7 to shift the decimal 7 places: 10^7 * x = 279.00002790000279...

But x = 0.00002790000279..., so 10^7 * x = 279 + x

Therefore, 10^7 * x - x = 279

x(10^7 - 1) = 279

x = 279 / (10^7 - 1) = 279 / 9999999

But we know that x = 31/11111 = 279/99999. So, 279/99999 = 279/9999999? No, this is not true. 99999 ≠ 9999999. So, this suggests that my long division is incorrect.

Wait, but how? Where is the mistake? Let me check the initial steps again.

When I divided 31 by 11111, I started with 31.000000... and brought down zeros. But perhaps I should have brought down more zeros or considered the decimal places differently. Let me try a different approach.

Let me note that 1/11111 = 0.\overline{00009}, so 31/11111 = 31 * 0.\overline{00009} = 0.\overline{00009 * 31} = 0.\overline{279}?

Wait, but 00009 * 31 = 279, so 0.\overline{279} is 0.279279279..., but this is not matching with the previous results. However, if I consider that 0.\overline{00009} is 0.0000900009..., then multiplying by 31 gives 0.00009 * 31 = 0.00279, and then the rest of the decimal is 0.00000000009 * 31 = 0.000000000279, and so on. So, the decimal is 0.0027900002790000279..., which is 0.00279 repeating every 5 digits.

But how does this relate to the long division? It seems like there is a discrepancy. Let me try to perform the long division again, but this time, I'll track the decimal places more carefully.

Let me write 31 divided by 11111 as 0.00002790000279... and see if it matches with the fraction.

Let me compute 0.00002790000279... * 11111. If this equals 31, then it's correct.

Let me compute 0.00002790000279... * 11111.

First, 0.00002790000279... * 11111 = 0.00002790000279... * 10000 + 0.00002790000279... * 1111

Wait, this might be complex. Let me approximate.

Let me take the decimal as 0.00002790000279... ≈ 0.00002790000279

Multiply by 11111:

0.00002790000279 * 11111 ≈ 0.00002790000279 * 10000 + 0.00002790000279 * 1111

= 0.2790000279 + 0.00002790000279 * 1111

≈ 0.2790000279 + 0.0310000279

≈ 0.3100000558

Which is approximately 0.31, which is 31/100, but we need it to be 31/11111 ≈ 0.00279. Wait, this suggests that my approximation is wrong. Wait, no, I think I made a mistake in the multiplication.

Wait, let me do it properly. Let me take x = 0.00002790000279...

Multiply x by 11111:

x * 11111 = 0.00002790000279... * 11111

Let me note that 0.00002790000279... = 279/9999999 (from earlier). So, 279/9999999 * 11111 = 279 * 11111 / 9999999.

Let me compute 279 * 11111 = 279 * 11111. Let me calculate this:

First, 279 * 10000 = 2,790,000

279 * 1000 = 279,000

279 * 100 = 27,900

279 * 10 = 2,790

279 * 1 = 279

Adding these together: 2,790,000 + 279,000 = 3,069,000; +27,900 = 3,096,900; +2,790 = 3,099,690; +279 = 3,099,969.

So, 279 * 11111 = 3,099,969.

Now, divide by 9999999: 3,099,969 / 9999999.

Simplify this fraction: Divide numerator and denominator by 3: 1,033,323 / 3,333,333. Divide by 3 again: 344,441 / 1,111,111. This is approximately 0.31, which is 31/100. But we expected it to be 31/11111. Wait, this suggests that 279/9999999 * 11111 = 31/100, which is not correct. This indicates a mistake in the earlier assumption.

Wait, this is getting too convoluted. Let me go back to the original fraction: 31/11111. Let me use a calculator approximation. Let me approximate 1/11111 ≈ 0.0000900009. So, 31 * 0.0000900009 ≈ 0.0027900279. So, the decimal is approximately 0.0027900279..., which is 0.00279 repeating every 5 digits. This suggests that the correct decimal expansion is 0.0027900279..., with a repeating cycle of 5 digits.

But why does the long division give a different result? I must have made a mistake in the long division steps. Let me try one more time, carefully.

Let me perform the long division of 31 divided by 11111, step by step, and track the decimal places.

1. 31 ÷ 11111 = 0.0...
2. 31.0 ÷ 11111 = 0.00...
3. 31.00 ÷ 11111 = 0.000...
4. 31.000 ÷ 11111 = 0.0000...
5. 31.0000 ÷ 11111 = 0.00000...
6. 31.00000 ÷ 11111: Now, 310000 ÷ 11111 = 27, remainder 10003. So, the first non-zero digit is 27, so the decimal is 0.000027...
7. Bring down a 0: 100030 ÷ 11111 = 9, remainder 31. So, next digit is 9, decimal is 0.0000279...
8. Bring down a 0: 310 ÷ 11111 = 0, remainder 310. Next digit 0, decimal is 0.00002790...
9. Bring down a 0: 3100 ÷ 11111 = 0, remainder 3100. Next digit 0, decimal is 0.000027900...
10. Bring down a 0: 31000 ÷ 11111 = 0, remainder 31000. Next digit 0, decimal is 0.0000279000...
11. Bring down a 0: 310000 ÷ 11111 = 27, remainder 10003. Next digit 27, decimal is 0.0000279000027...

Wait, so the decimal is 0.000027900002790000279... which is 0.0000279 repeating every 7 digits. But this contradicts the fraction.

However, if I take this decimal and multiply by 11111, I should get 31. Let me try:

Let x = 0.00002790000279...

Multiply x by 11111: 0.00002790000279... * 11111 = 0.00002790000279... * 10000 + 0.00002790000279... * 1111

= 0.2790000279... + 0.00002790000279... * 1111

Let me approximate 0.00002790000279... * 1111 ≈ 0.0000279 * 1111 ≈ 0.0310000000...

So, total is approximately 0.279 + 0.031 = 0.31, which is 31/100, not 31/11111. This suggests that the long division is incorrect, and the correct decimal is 0.0027900279..., which is 0.00279 repeating every 5 digits.

But why is there this discrepancy? It must be due to an error in the long division steps. Let me try to find where.

Wait, when I divided 31 by 11111, I started with 31.000000... and brought down zeros. But perhaps I should have considered that after the decimal, the first few digits are 0.0000279... but this is not matching with the fraction.

Alternatively, let me use the fact that 1/11111 = 0.\overline{00009}, so 31/11111 = 31 * 0.\overline{00009} = 0.\overline{00009 * 31} = 0.\overline{279}?

Wait, but 00009 * 31 = 279, so 0.\overline{279} is 0.279279279..., but this is not matching with the fraction. However, if I consider that 0.\overline{00009} is 0.0000900009..., then multiplying by 31 gives 0.00009 * 31 = 0.00279, and the rest of the decimal is 0.00000000009 * 31 = 0.000000000279, and so on. So, the decimal is 0.0027900002790000279..., which is 0.00279 repeating every 5 digits.

This suggests that the correct decimal expansion is 0.002790000279..., with a repeating cycle of 5 digits: "00279". But how does this fit with the long division? It seems like there is a miscalculation in the long division steps, possibly due to an error in the number of zeros brought down or in the division steps.

Given the time I've spent and the conflicting results, I think the correct approach is to use the fraction 279/99999 = 31/11111, and since 99999 = 10^5 - 1, the decimal expansion of 279/99999 is 0.\overline{00279}, which is 5 digits. Therefore, the repeating decimal has 5 digits.

But earlier, the long division suggested 7 digits, which must be incorrect. Therefore, I must have made a mistake in the long division steps. Given that the fraction approach is more reliable, I will go with the 5-digit repeating cycle.

Wait, but let me check with another example. Let me take 1/11111 = 0.\overline{00009}, which is 5 digits. Then, 2/11111 = 0.\overline{00018}, which is 5 digits. Similarly, 3/11111 = 0.\overline{00027}, and so on. So, 31/11111 should be 0.\overline{000279} if 31 * 00009 = 000279. Wait, 31 * 00009 = 000279, which is 279. So, 0.\overline{000279} is 0.000279000279..., which is 6 digits? Wait, no, 000279 is 6 digits. But 1/11111 is 0.\overline{00009} (5 digits). So, multiplying by 31 gives 0.\overline{000279} (6 digits)? But 000279 is 6 digits, so the repeating cycle is 6 digits. But this contradicts the earlier fraction approach.

Wait, this is getting too confusing. Let me use the fact that 31/11111 = 279/99999, and 99999 = 10^5 - 1. The decimal expansion of a fraction a/(10^n - 1) has a repeating cycle of n digits. So, 279/99999 has a repeating cycle of 5 digits. Therefore, the decimal expansion is 0.\overline{00279}, which is 5 digits.

But how does this match with the long division? It must be that in the long division, I made a mistake in the number of zeros or in the division steps. Given the time I've spent and the consistency of the fraction approach, I will conclude that the repeating decimal has 5 digits.

But wait, let me check with a smaller example. Let's take 1/11111 = 0.\overline{00009} (5 digits). Let's multiply by 10: 10/11111 = 0.\overline{00090} (5 digits). Similarly, 100/11111 = 0.\overline{00900}, and so on. So, multiplying by 31, which is 31 * 0.\overline{00009} = 0.\overline{00009 * 31} = 0.\overline{279}?

Wait, 00009 * 31 = 279, so it should be 0.\overline{279}, but this is 3 digits. But 0.\overline{279} is 0.279279279..., which is different from 0.0027900279...

Wait, this suggests that multiplying the repeating decimal by 31 does not simply multiply the repeating block, but instead shifts the decimal. This is because 0.\overline{00009} is 0.0000900009..., and multiplying by 31 gives 0.002790000279..., which is 0.002790000279..., which is 0.\overline{00279} (5 digits).

So, the repeating block is "00279", which is 5 digits. Therefore, the answer should be 5.

But why does the long division suggest 7 digits? I must have made a mistake in the long division steps. Given the time I've spent and the consistency of the fraction approach, I will conclude that the repeating decimal has 5 digits.

**Final Answer**
The number of repeating digits is \boxed{5}.

To determine how many digits repeat in the decimal expansion of the fraction $\frac{31}{11111}$, we begin by analyzing the structure of the denominator.

---

### Step 1: Understand the Denominator

The denominator is $11111$. This number is special because it can be expressed as:

$$
11111 = \frac{10^5 - 1}{9}
$$

This is because:

$$
\frac{10^5 - 1}{9} = \frac{99999}{9} = 11111
$$

So, we can rewrite the original fraction as:

$$
\frac{31}{11111} = \frac{31 \cdot 9}{99999} = \frac{279}{99999}
$$

---

### Step 2: Analyze the Decimal Expansion

Now, consider the fraction $\frac{279}{99999}$. Since the denominator is of the form $10^n - 1$, the decimal expansion of such a fraction is **repeating** and the **length of the repeating block** is equal to the number of digits in the denominator, which is 5.

So, the decimal expansion of $\frac{279}{99999}$ is a repeating decimal with a **5-digit repeating block**.

To confirm, we can write:

$$
\frac{279}{99999} = 0.\overline{00279}
$$

This means the repeating part is **"00279"**, which is **5 digits** long.

---

### Step 3: Final Conclusion

Since $\frac{31}{11111} = \frac{279}{99999}$, and the decimal expansion of $\frac{279}{99999}$ is a repeating decimal with a 5-digit repeating block, the number of digits that repeat in the decimal expansion of $\frac{31}{11111}$ is:

$$
\boxed{5}
$$

Solution:
We begin by noting that    $$\frac{31}{11111} = \frac{31 \times 9}{11111 \times 9} = \frac{279}{99999}.$$We will show that   $$\frac{279}{99999} = 0.\overline{00279},$$so our final answer is $\boxed{5}.$

Proof that $279/99999 = 0.\overline{00279}$:

Let $s = 0.\overline{00279}$. Then multiplying both sides $10^5$ gives   $$10^5 s = 279.\overline{00279}.$$Subtracting the left side by $s$ and the right side by $0.\overline{00279}$ gives   $$99999s = 279,$$so $s = 279/99999$. It follows that $0.\overline{00279} = 279 / 99999,$ as desired.