Question

Look at several examples of rational numbers in the form p/q ( q ≠ 0 ),where p and q 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 consider the examples of the form p/q that are terminating decimals.

5/2 = 2.5 , 5/4 = 1.25, 2/5 = 0.4, 2/10 = 0.2, = 5/16 = 0.3125

We can observe that the denominators of the above rational numbers have powers of 2, 5 or both.

Therefore, we can conclude that the property, which q must satisfy in p/q, so that the rational number p/q is a terminating decimal is that q must have powers of 2, 5 or both.