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# The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form $\frac{p}{q}$, what can you say about the prime factors of $q$?

**(i)** $43.123456789$

**(ii)** $0.120120012000120000$....

**(iii)** $43.\overline{123456789}$.

To do:

Here, we have to determine the nature of the prime factorisation of the denominator of the given rational numbers when expressed in $\frac{p}{q}$ form.

Solution:

(i) $43.123456789$ has a terminating decimal expansion.

This implies it is a rational number of the form $\frac{p}{q}$ and $q$ is of the form $2^m \times 5^n$, where $p$ and $q$ are non-negative integers. The prime factors of the denominator of the given rational number are $2$ and $5$.

(ii) $0.120120012000120000$.... has a non-terminating decimal expansion.

The denominator of the given rational number has factors other than $2$ or $5$.

Therefore, the given real number is not rational.

(iii) $43.\overline{123456789}$ has a non-terminating decimal expansion.

Therefore, the denominator of the given rational number has factors other than $2$ or $5$.

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