You know that `1/7=0. bar 142857`Can you predict what the decimal expansion of `2/7,3/7,4/7,5/7,6/7`are, without actually doing the long division? If so, how? [Hint: Study the remainders while finding the value of `1/7` carefully.]

Explanation

To find the decimal expansions of \( \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7} \) based on the known decimal expansion of \( \frac{1}{7} = 0.\overline{142857} \), we can use the pattern observed in the remainders when performing long division for \( \frac{1}{7} \).

### Step-by-Step Solution:

1. **Understanding \( \frac{1}{7} \)**:
We know that \( \frac{1}{7} = 0.\overline{142857} \). The repeating decimal "142857" has a length of 6 digits.

2. **Finding \( \frac{2}{7} \)**:
...