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?
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 2m x 5n ; m = 0,1,2,3,............, n = 0,1,2,3,.............
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