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

The correct option is A

Rational Number, Prime factors of q will be of form $2^n{.5}^m$ , $m$ and $n$ are whole numbers.

It is a rational number because the decimal expansion is non-terminating but repeating. Therefore, it can be expressed in $\frac{p}{q}$ form where factors of q are of the form $2^n{.5}^m$ where n and m are whole numbers.