Question

Without actually performing the long division state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

$\frac{13}{3125}$.

Answer 1

If the denominator has factors of the form of $2^m\times 5^n$ then it has a terminating decimal expansion.

If the denominator has factors other than $2$ and $5$ then it has a non-terminating decimal expansion.

Now, factorizing the denominator.

$$\begin{gathered} 3125=5\times 5\times 5\times 5\times 5 \\ =5^5 \end{gathered}$$

As the denominator has only $5$ as its factor.

Hence, $\frac{13}{3125}$ has a terminating decimal expansion.