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(B) rational number $\frac{p}{q}$ has a terminating decimal expansion if the prime factorization of the denominator $q$ is of the form $2^n 	imes 5^m

Vedclass · Accepted Answer

$\frac{17}{248}$

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(B) rational number $\frac{p}{q}$ has a terminating decimal expansion if the prime factorization of the denominator $q$ is of the form $2^n 	imes 5^m$,where $n$ and $m$ are non-negative integers.$1$. For $\frac{17}{32}$: $32 = 2^5$. Since it is in the form $2^n 	imes 5^m$,it has a terminating decimal expansion.$2$. For $\frac{17}{248}$: $248 = 8 	imes 31 = 2^3 	imes 31^1$. Since there is a factor other than $2$ or $5$ (i.e.,$31$),it has a non-terminating repeating decimal expansion.$3$. For $\frac{17}{160}$: $160 = 16 	imes 10 = 2^4 	imes 2 	imes 5 = 2^5 	imes 5^1$. Since it is in the form $2^n 	imes 5^m$,it has a terminating decimal expansion.Therefore,$\frac{17}{248}$ has a non-terminating repeating decimal expansion.

Which of the following rational numbers has a non-terminating repeating decimal expansion? $(A)$ $\frac{17}{32}$,$(B)$ $\frac{17}{248}$,$(C)$ $\frac{17}{160}$

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