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?

Answer 1

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.