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Look at several examples of rational numbers in the form p/q (q ≠ 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?

6. Look at several examples of rational numbers in the form $\dpi{100} \frac{p}{q}$ (q ≠ 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?

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We can observe that when q is 2, 4, 5, 8, 10… then the decimal expansion is terminating. For example:

$\frac{3}{2}= 1.5$, denominator $q = 2^1$

$\frac{8}{5}= 1.6$, denominator $q = 5^1$

$\frac{15}{10} = 1.5$ , denominator $q =10=2\times 5= 2^1 , 5^1$

Therefore,

It can be observed that the terminating decimal can be obtained in a condition where prime factorization of the denominator of the given fractions has the power of 2 only or 5 only or both.

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