What can be the maximum number of digits in the repeating block of digits in the expansion of `(5)/(7)` ?

Answer

To find the maximum number of digits in the repeating block of digits in the decimal expansion of \( \frac{5}{7} \), we can follow these steps:

### Step 1: Understand the Decimal Expansion
When we divide \( 5 \) by \( 7 \), we want to find out how many digits repeat in the decimal representation.

### Step 2: Perform the Division
We will perform the long division of \( 5 \) by \( 7 \):
1. Since \( 5 < 7 \), we start by adding a decimal point and a zero, making it \( 50 \).
2. \( 7 \) goes into \( 50 \) seven times (since \( 7 \times 7 = 49 \)).
3. Subtract \( 49 \) from \( 50 \), which leaves us with \( 1 \).
4. Bring down another \( 0 \) to make it \( 10 \).
5. \( 7 \) goes into \( 10 \) once (since \( 7 \times 1 = 7 \)).
6. Subtract \( 7 \) from \( 10 \), which leaves us with \( 3 \).
7. Bring down another \( 0 \) to make it \( 30 \).
8. \( 7 \) goes into \( 30 \) four times (since \( 7 \times 4 = 28 \)).
9. Subtract \( 28 \) from \( 30 \), which leaves us with \( 2 \).
10. Bring down another \( 0 \) to make it \( 20 \).
11. \( 7 \) goes into \( 20 \) two times (since \( 7 \times 2 = 14 \)).
12. Subtract \( 14 \) from \( 20 \), which leaves us with \( 6 \).
13. Bring down another \( 0 \) to make it \( 60 \).
14. \( 7 \) goes into \( 60 \) eight times (since \( 7 \times 8 = 56 \)).
15. Subtract \( 56 \) from \( 60 \), which leaves us with \( 4 \).
16. Bring down another \( 0 \) to make it \( 40 \).
17. \( 7 \) goes into \( 40 \) five times (since \( 7 \times 5 = 35 \)).
18. Subtract \( 35 \) from \( 40 \), which leaves us with \( 5 \).

At this point, we see that we are back to \( 5 \), which means the decimal will start repeating.

### Step 3: Identify the Repeating Block
The decimal expansion of \( \frac{5}{7} \) is \( 0.\overline{714285} \). The repeating block is \( 714285 \).

### Step 4: Count the Digits in the Repeating Block
The repeating block \( 714285 \) consists of \( 6 \) digits.

### Conclusion
Thus, the maximum number of digits in the repeating block of digits in the expansion of \( \frac{5}{7} \) is \( 6 \).

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