Question:
What can you say about the prime factorisations of the denominators of the following rationals:
(i) 43.123456789
(ii) $43 . \overline{123456789}$
(iii) $27 . \overline{142857}$
(iv) 0.120120012000120000 ...
Solution:
(i) Since 43.123456789 has terminating decimal expansion.
So, its denominator is of the form 2m × 5n, where m, n are non-negative integers.
(ii) Since $43.123456789$ has non-terminating decimal expansion.
So, its denominator has factors other than 2 or 5 .
(iii) Since $27 . \overline{142857}$ has non-terminating decimal expansion.
So, its denominator has factors other than 2 or 5 .
(iv) Since $0.120120012000120000 \ldots$ has non-terminating decimal expansion.
So, its denominator has factors other than 2 or 5 .