Question

Look at several examples of rational number in the form $\frac{p}{q}(q\ne 0)$ , 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 ?

Answer 1

Defining properties of $q$:

Any number can be represented by $\frac{p}{q}$ and $q\ne 0$ is called a rational number. Numbers that cannot be represented by $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$ is known as an irrational number.
$q$must satisfy the condition that if prime factors of $q$ have only powers of $2$or power of $5$or both, then the rational numbers always have a terminating decimal expansion, i.e., $2^m\times 5^n$, where $m=1,2,3,\cdots$ or $n=1,2,3......$
Example:
$\frac{1}{2}=0.5$, denominator $q=2^1$
$\frac{7}{8}=0.875$, denominator $q=2^3$
$\frac{4}{5}=0.8$, denominator $q=5^1$
Hence, $q$ should be in the format of $2^{m}x{5}^n$,where $n$ and $m$ are natural numbers.