Question

Decide whether the real number with decimal expansion as $125.\overline{987654321}$ is rational or not. If rational, and of the form $\frac{p}{q}$, what can you say about the prime factors of $q$?

• A. Rational Number, Prime factors of q will be of form $2^n{.5}^m$ , $m$ and $n$ are whole numbers.
• B. Not rational Number
• C. Rational Number, prime factor cannot be represented in the form $2^n{.5}^m$, $m$ and $n$ are whole numbers.
• D. Rational Number, Prime factor of q will be 3.

Answer 1

Answer: (A)

Given number:

$125.\overline{987654321}$

It is a rational number because the decimal expansion is non-terminating but repeating. Therefore, it can be expressed in $\frac{p}{q}$.

Now, lets convert into $\frac{p}{q}$ form.

Let,

$x=125.\overline{987654321}$

$\Rightarrow x=\mathrm{125.987654321987654321987654321...}$ --(1)

Since, $9$ digits are recurring, multiply by ${10}^9$ on both sides,

$\Rightarrow {10}^9\times x={10}^9\times \mathrm{125.987654321987654321987654321...}$

$\Rightarrow {10}^9\times x=\mathrm{125987654321.987654321987654321...}$---(2)

Lets do {2)-(1)

${10}^9\times x-x=\mathrm{125987654321.987654321987654321...}-\mathrm{125.987654321987654321987654321...}$

$\Rightarrow x({10}^9-1)=125987654196$

$\Rightarrow x=\frac{125987654196}{{10}^9-1}$

Here, $q={10}^9-1$

Since, ${10}^9=2^9\times 5^9$

$\Rightarrow {10}^9-1$ is not in the form of $2^9\times 5^9$.

So it is non terminating and recurring decimal

Conclusion:

In rational Number $\frac{p}{q}$, Prime factors of $q$ will be of form $2^n{.5}^m$ , $m$ and $n$ are whole numbers.

Hence, Option A is correct.