Question

Look at several examples of rational numbers 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 (expansions). Can you guess what property $q$ must satisfy?

Answer 1

We observe that when $q$ is 2, 4, 5, 8, 10... then the decimal expansion is terminating. For 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 $q=5^1$

$\frac{50}{20}=2.5$, denominator $q=2^2\times 5^1$

$\frac{450}{100}=4.5$, denominator $q=(2\times 5{)}^2$

We observed that, If a rational number is a terminating decimal then its denominator can be express in the form of either $2^n$ or $5^n$ or $2^n\times 5^m$ or $(2\times 5{)}^n$.

Where, $m,n$ are natural numbers.