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# 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.