Look at several examples of rational numbers in the form $\frac{p}{q}$ ($q ≠ 0$), where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal representations. Can you guess what property $q$ must satisfy?

To do:

We have to guess the property that $q$ has to satisfy for the given condition.

Solution:

Few rational numbers which have a terminating decimal representation are:

$\frac{1}{2}=0.5$

$\frac{1}{4}=0.25$

$\frac{1}{5}=0.2$

$\frac{1}{10}=0.1$

We can observe that,

If we have a rational number $\frac{p}{q}$, where $p$ and $q$ are co-primes and the prime factorization of $q$ is of the form $2^n.5^m$, where $n$ and $m$ are non-negative integers, then $\frac{p}{q}$ has a terminating expansion.

The property is if the denominator has factors 2 or 5 or both, the decimal representation will be terminating.