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# You know that $$\frac{1}{7} = 0\overline{.142857.}$$ Can you predict what the decimal expansions of $$\frac{2}{7} , \frac{3}{7} , \frac{4}{7} , \frac{5}{7} , \frac{6}{7}$$ are, without actually doing the long division. If so, how?[Hint Study the remainders while finding the value $$\frac{1}{7}$$ of carefully.]

We have,
$$\frac{1}{7} = 0\overline{.142857}$$

So we can say that,
$$\because \frac{2}{7} = 2 × \frac{1}{7}$$
$$\Rightarrow \frac{2}{7} = 2 × 0\overline{.142857}$$
$$\Rightarrow \frac{2}{7} = 0\overline{.285714}$$

$$\because \frac{3}{7} = 3 × \frac{1}{7}$$
$$\Rightarrow \frac{3}{7} = 3 × 0\overline{.142857}$$
$$\Rightarrow \frac{3}{7} = 0\overline{.428571}$$

$$\because \frac{4}{7} = 4 × \frac{1}{7}$$
$$\Rightarrow \frac{4}{7} = 4 × 0\overline{.142857}$$
$$\Rightarrow \frac{4}{7} = 0\overline{.571428}$$

$$\because \frac{5}{7} = 5 × \frac{1}{7}$$
$$\Rightarrow \frac{5}{7} = 5 × 0\overline{.142857}$$
$$\Rightarrow \frac{5}{7} = 0\overline{.714285}$$

$$\because \frac{6}{7} = 6 × \frac{1}{7}$$
$$\Rightarrow \frac{6}{7} = 6 ×0 \overline{.142857}$$
$$\Rightarrow \frac{6}{7} = 0\overline{.857142}$$