Question

Question 6
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 $q=5^1$

We can observe that terminating decimal may be obtained in the situation where prime factorization of the denominator of the given fractions has the power of 2 only or 5 only or both.