The following real number has a decimal expansion as given below. Decide whether it is rational or not. If it is rational,and of the form $\frac{p}{q}$,what can you say about the prime factors of $q$?
$43. \overline{123456789}$

(N/A) The given number is $43. \overline{123456789}$.
Since the decimal expansion is non-terminating and repeating (recurring),the given number is a rational number.
$A$ rational number has a terminating decimal expansion if and only if the prime factorization of its denominator $q$ is of the form $2^m \times 5^n$,where $m$ and $n$ are non-negative integers.
Since this decimal expansion is non-terminating repeating,the denominator $q$ must have prime factors other than $2$ or $5$.