Q.2. You know that $\frac{1}{7}$ = 0.$\stackrel{‾}{142857}$. Can you predict what the decimal expansion 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 of $\frac{1}{7}$ carefully.]

Solution:

Yes. We can be done this by simply multiplying the RHS by the numerator, i.e., 2, 3, 4 etc.

$\frac{2}{7}=2×0.\stackrel{‾}{142857}=0.\stackrel{‾}{285714}$

$\frac{3}{7}=3×0.\stackrel{‾}{142857}=0.\stackrel{‾}{428571}$

$\frac{4}{7}=4×0.\stackrel{‾}{142857}=0.\stackrel{‾}{571428}$

$\frac{5}{7}=5×0.\stackrel{‾}{142857}=0.\stackrel{‾}{714285}$

$\frac{6}{7}=6×0.\stackrel{‾}{142857}=0.\stackrel{‾}{857142}$