Without actually performing the long division, state whether the following rational number has terminating or non-terminating repeating (recurring) decimal expansion: $$\dfrac{29}{343}$$
Given rational number is $$\dfrac{29}{343}$$
$$\dfrac{p}{q}$$ is terminating if
a) $$p$$ and $$q$$ are co-prime and
b) $$q$$ is of the form of $$2^{n}5^{m}$$ where $$n$$ and $$m$$ are non-negative integers.
Firstly we check co-prime.
$$29=29*1$$
$$343=7*7*7$$
$$\Rightarrow 29$$ and $$343$$ have no common factors.
Therefore, $$29$$ and $$343$$ are co-prime.
Now, we have to check that $$q$$ is in the form of $$2^{n}5^{m}$$.
$$343=7^{3}$$
So, the denominator is not of the form $$2^{n}5^{m}$$.
Thus, $$\frac{29}{343}$$ is a non-terminating repeating decimal.
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Which of the following fractions will terminate when expressed as a decimal? (Choose all that apply.)
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