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 predict the decimal expansions of \( \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7} \) based on the known value of \( \frac{1}{7} = 0.\overline{142857} \), we can use the property of multiplying the decimal expansion of \( \frac{1}{7} \) by integers from 2 to 6.

### Step-by-Step Solution:

1. **Identify the Decimal Expansion of \( \frac{1}{7} \)**:
\[ \frac{1}{7} = 0.\overline{142857} \]

2. **Calculate \( \frac{2}{7} \)**:
\[ \frac{2}{7} = 2 \times \frac{1}{7} = 2 \times 0.\overline{142857} \]
- Multiply \( 0.142857 \) by 2:
\[ 0.285714 \text{ (the decimal repeats)} \]
- Thus,
\[ \frac{2}{7} = 0.\overline{285714} \]

3. **Calculate \( \frac{3}{7} \)**:
\[ \frac{3}{7} = 3 \times \frac{1}{7} = 3 \times 0.\overline{142857} \]
- Multiply \( 0.142857 \) by 3:
\[ 0.428571 \text{ (the decimal repeats)} \]
- Thus,
\[ \frac{3}{7} = 0.\overline{428571} \]

4. **Calculate \( \frac{4}{7} \)**:
\[ \frac{4}{7} = 4 \times \frac{1}{7} = 4 \times 0.\overline{142857} \]
- Multiply \( 0.142857 \) by 4:
\[ 0.571428 \text{ (the decimal repeats)} \]
- Thus,
\[ \frac{4}{7} = 0.\overline{571428} \]

5. **Calculate \( \frac{5}{7} \)**:
\[ \frac{5}{7} = 5 \times \frac{1}{7} = 5 \times 0.\overline{142857} \]
- Multiply \( 0.142857 \) by 5:
\[ 0.714285 \text{ (the decimal repeats)} \]
- Thus,
\[ \frac{5}{7} = 0.\overline{714285} \]

6. **Calculate \( \frac{6}{7} \)**:
\[ \frac{6}{7} = 6 \times \frac{1}{7} = 6 \times 0.\overline{142857} \]
- Multiply \( 0.142857 \) by 6:
\[ 0.857142 \text{ (the decimal repeats)} \]
- Thus,
\[ \frac{6}{7} = 0.\overline{857142} \]

### Final Results:
- \( \frac{2}{7} = 0.\overline{285714} \)
- \( \frac{3}{7} = 0.\overline{428571} \)
- \( \frac{4}{7} = 0.\overline{571428} \)
- \( \frac{5}{7} = 0.\overline{714285} \)
- \( \frac{6}{7} = 0.\overline{857142} \)