2. You know that 1/7 = 0.142857. Can you predict what the decimal expansions 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.]
Solution: We have the decimal expansion of 1/7 = 0.142587 Let's find the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7. To proceed with this, let's observe the pattern of remainders and the digits of the quotient in the long division of 1/7 as shown below. Summary: Given the decimal expansion of 1/7 = 0.142587, the decimal expansions of 2/7, 3/7, 4/7, 5/7, and 6/7 are 0.285714, 0.428571, 0.571428, 0.714285, and 0.857142 respectively.
