Find the six-digit number $abcdef$ such that

Question

0.\overline{abcdef} = \frac{2}{17},
without a calculator.

Answer 1

To convert the repeating decimal $0.\overline{abcdef}$ to a fraction, we can use the formula for a geometric series.
Let $x = 0.\overline{abcdef}$. Then, multiplying both sides by 1000000 (since there are 6 digits in the repeating block), we have $1000000x = abcdef.\overline{abcdef}$. Subtracting the original equation from the second equation, we get:
$$999999x = abcdef$$.
We are given that this fraction is equal to $\frac{2}{17}$. Thus, we have:
$$\frac{999999x}{1000000}=\frac{2}{17}$$
$$x=\frac{2 \cdot 1000}{17 \cdot 999}$$
$$x=\frac{2000}{17001} = \frac{2000}{17 \cdot 1001}$$
$$x=\frac{2000}{17} \cdot \frac{1}{1001}$$
$$x = \frac{2000}{17} \cdot \frac{1}{1001}=\frac{2000}{17} \cdot \frac{1001}{1001}=\frac{2000000}{17017}$$.

Therefore, the six-digit number is $\boxed{117647}$.