When we divide 3 by 7 we get 0.428571428571.... therefore the first 6 numbers are recurring, so the answer is 6.

Maximum Number of Entries in the Repeating String of Remainders

To find the maximum number of entries in the repeating string of remainders in the decimal expansion of 3/7, we need to perform the division and observe the pattern of remainders that emerge.

Division of 3 by 7
When we divide 3 by 7, the decimal representation is 0.42857142857...

Repeating String of Remainders
The repeating string of remainders is the pattern that emerges after the decimal point. In this case, the repeating string is 428571.

Explanation
To understand why the repeating string of remainders occurs, let's go through the steps of the division process.

1. We start by dividing 3 by 7. The quotient is 0 and the remainder is 3.
2. We multiply the remainder (3) by 10 and divide by 7. This gives us a quotient of 4 and a remainder of 2.
3. Again, we multiply the remainder (2) by 10 and divide by 7. This gives us a quotient of 2 and a remainder of 6.
4. We repeat this process, multiplying the remainder by 10 and dividing by 7, until we obtain a remainder of 3 again. At this point, the pattern repeats.

The repeating string of remainders is obtained by noting down the remainders at each step of the division process. In this case, the remainders are 3, 2, 6, 4, 5, 1, and then the pattern repeats with 3, 2, 6, and so on.

Maximum Number of Entries
To find the maximum number of entries in the repeating string of remainders, we need to determine the length of the pattern before it starts repeating. In this case, the length of the repeating string is 6 (428571), so the maximum number of entries is 6.

Conclusion
The maximum number of entries in the repeating string of remainders in the decimal expansion of 3/7 is 6. The repeating string is 428571, obtained by observing the remainders at each step of the division process.