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Question
You know that `1/7=0.bar142857.` 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
Given `1/7 = 0.bar142857`
The decimal expansions of `2/7, 3/7, 4/7, 5/7, 6/7` without long division are as follows:
`2/7=2xx1/7=2xx0.bar142857=0.bar285714`
`3/7=3xx1/7=3xx0.bar142857=0.bar428571`
`4/7=4xx1/7=4xx0.bar142857=0.bar571428`
`5/7=5xx1/7=5xx0.bar142857=0.bar714285`
`6/7=6xx1/7=6xx0.bar142857=0.bar857142`
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