Question

Given that $\dfrac{1}{7}=0.\overline{142857}$ which is a repeating decimal having six different digits. If x is the sum of such first three positive integers n such that $\dfrac{1}{n}=0.\overline{abcdef}$, where a, b, c, d, e and f are different digits, then the value of x is
(a) 20
(b) 21
(c) 41
(d) 42

Answer 1

Hint: First, before proceeding for this, we must know the definition and concept of the non-terminating fraction in which fraction doesn’t gets terminated and continues till infinity and in non-terminating fractions, there are some fractions which give repetition of particular digits only. Then, we are supposed to find the two more similar fractions which are non-terminating and also give six digits as repetitions where the bar represents the repetition of the digits. Then, we need to find the value of x which is sum of first three numbers which gives non-terminating repetition decimals.

Complete step by step answer:
In this question, we are supposed to find the value of x where $\dfrac{1}{7}=0.\overline{142857}$ which is a repeating decimal having six different digits. If x is the sum of such first three positive integers n such that $\dfrac{1}{n}=0.\overline{abcdef}$, where a, b, c, d, e, and f are different digits.
So, before proceeding for this, we must know the definition and concept of the non-terminating fraction in which fraction doesn’t get terminated and continues till infinity.
Also, we know that in non-terminating fractions there are some fractions that gives repetition of particular digits only.
Here, in this question, we are given with the first fraction as:
$\dfrac{1}{7}=0.\overline{142857}$
Now, we are supposed to find the two more similar fractions which are non-terminating and also give six digits as repetitions where bar represents the repetition of the digits.
Usually, the division with the prime number gives the non-terminating decimals.
So, the other number is 13 and the required fraction is as:
$\dfrac{1}{13}=0.\overline{076923}$
Similarly, the next number which gives the six repeated digits as a decimal is as:
$\dfrac{1}{21}=0.\overline{047619}$
So, we get the required number as 7, 13, and 21.
Now, we need to find the value of x which is the sum of the above-mentioned number.
So, we get the value of x as:
x=7+13+21
$\Rightarrow x=41$
So, we get the value of x as 41.
Hence, the option (c) is correct.

Note:
Now, to solve these types of questions we need to know some of the basics of the fraction so that questions can be solved easily. Moreover, non terminating fractions have a bar on some number which represent that those numbers are repeated in the fraction.