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Question
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
Let us look at the decimal expansion of the following terminating rational numbers:
`3/2` = `(3 xx 5)/(2 xx 5) = 15/10 = 1.5` [Denominator = 2 = 21]
`1/5` = `(1 xx 2)/(5 xx 2) = 2/10 = 0.2` [Denominator = 5 = 51]
`7/8` = `(7 xx 125)/(8 xx 125) = 875/1000 = 0.875` [Denominator = 8 = 23]
`8/125` = `(8 xx 8)/(125 xx 8) = 64/1000 = 0.064` [Denominator = 125 = 53]
`13/20` = `(13 xx 5)/(20 xx 5) = 65/100 = 0.65` [Denominator = 20 = 22 = 51]
`17/16` = `(17 xx 625)/(16 xx 625) = 10625/10000 = 1.0625` [Denominator = 16 = 24]
We observe that the prime factorisation of q (i.e. denominator) has only powers of 2 or powers of 5 or powers of both.
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