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?

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#### Solution

The property that **q** must satisfy in order that the rational numbers in the form p/q, where **p** and **q** are integers with no common factors other than 1, have terminating decimal representation (expansions) is that the prime factorization of **q** has only powers of 2 or powers of 5 or both, i.e., **q** must be of the form 2^{m} x 5^{n} ; **m** = 0,1,2,3,............, **n** = 0,1,2,3,.............

Concept: Real Numbers and Their Decimal Expansions

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