6. Look at several examples of rational numbers in the form \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?

Answers (1)

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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