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A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the form $\frac{p}{q}$? Give reasons.
Given:
The given decimal expansion of a rational number is 327.7081.
To do:
Here, we have to determine the nature of the prime factors of the denominator of the given rational number when expressed in $\frac{p}{q}$ form.
Solution:
$327.7081.$ 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 $5$.
Proof:
$327.7081$ can be written as,
$327.7081=\frac{3277081}{10000}$
$=\frac{3277081}{(10)^4}$
$=\frac{3277081}{(2\times5)^4}$
$=\frac{3277081}{2^4\times5^4}$
The denominator is of the form $2^m \times 5^n$, where $p$ and $q$ are non-negative integers. Therefore, the prime factors of $q$ has only factors of $2$ and $5$.