Subjective Type

Without actually performing the long division, state whether the following rational number have terminating or non-terminating repeating (recurring) decimal expansion: $$\frac{17}{8}$$.

Solution

Given, rational number is $$\frac{17}{8}$$.
We know, $$\frac{p}{q}$$ is terminating if:
a) $$p$$ and $$q$$ are co-prime and
b) $$q$$ is of the form of $$2^{n}\times5^{m}$$, where $$n$$ and $$m$$ are non-negative integers.
Firstly, we check whether the numerator and denominator are co-prime,
$$17=17\times1$$
$$8=2\times2\times2$$
$$\Rightarrow 17$$ and $$8$$ have no common factors.
Therefore, $$17$$ and $$8$$ are co-prime.
Now, we check whether $$q$$ is in the form of $$2^{n}\times5^{m}$$,
denominator $$8=2^{3}$$ $$=1\times2^{3}$$ $$=5^{0}\times2^{3}$$
Since the denominator is of the form $$2^{n}\times5^{m}$$ where $$n=3$$ and $$m=0$$,
thus, $$\frac{17}{8}$$ has a terminating decimal expansion.

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