6.Look at several examples of rational numbers in the form p/q (\(q \ne 0\)). 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?

Can you guess what property q must satisfy?

Considering some rational numbers in the form p/q (q?0) with no common factors other than 1 and having terminating decimal representations(expansions).

we can say, the various such rational numbers are 1\2, 7/125, 1/4, 19/20, etc.

\(1/2 = (1 × 5)/(2 × 5) = 5/10 = 0.5.\)

\(7/125 = (7 × 8)/(125 × 8) = 56/1000 = 0.056\)

\(1/4 = (1 × 25)/(25 × 4) = 25/100 = 0.25\)

\(19/25 = (19 × 4)/(25 × 4) = 76/100 = 0.76 \)

In all the cases mentioned above, we think of the natural number which when multiplied by their respective denominators gives 10 or a power of 10.

Thus, we find that, the decimal expansion of above numbers are terminating.

Along with, we see that the denominator of above numbers i.e. q is in the form has only powers of 2 or power of 5 or both of them.