Look at several examples of rational numbers in the form \(\frac{p}{q} \) (\(q \neq 0\)). 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?

Considering some rational numbers in the form \(\frac{p}{q} \)( \(q \neq 0\) )with no common factors other than 1 and having terminating decimal representations(expansions).


we can say, the various such rational numbers are \(\frac{1}{2} , \frac{7}{125} , \frac{1}{4} , \frac{19}{20} \), etc.

\(\frac{1}{2} = \frac{(1 × 5)}{(2 × 5)} = \frac{5}{10} = 0.5.\)

\(\frac{7}{125} =\frac{(7 × 8)}{(125 × 8)} = \frac{56}{1000} = 0.056\)

\(\frac{1}{4} = \frac{(1 × 25)}{(25 × 4)} = \frac{25}{100} = 0.25\)

\(\frac{19}{25} = \frac{(19 × 4)}{(25 × 4)} = \frac{76}{100} = 0.76 \)

In all the cases mentioned above, we think of the natural number which when multiplied by their respective denominators gives 10 or a power of 10.

Thus, we find that, the decimal expansion of above numbers are terminating.

Along with, we see that the denominator of above numbers i.e. q is in the form has only powers of 2 or power of 5 or both of them.