What can you say about the prime factorization of the denominators of

1.	What can you say about the prime factorization of the denominators of the following rational:(i) 43.123456789(ii) \(43.\overline{123456789}\)(iii) \(27.\overline{142857}\)(iv) 0.120120012000120000….
Answer» (i) 43.123456789 Since 43.123456789 has terminating decimal expansion. Hence, its denominator is of the form 2^m x 5ⁿ, where m, n are non-negative integers. (ii) \(43.\overline{123456789}\) Since the given rational has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5. (iii) \(27.\overline{142857}\) Since the given rational number has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5. (iv) 0.120120012000120000…. Since 0.120120012000120000…. has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.

What can you say about the prime factorization of the denominators of the following rational:(i) 43.123456789(ii) \(43.\overline{123456789}\)(iii) \(27.\overline{142857}\)(iv) 0.120120012000120000….

Answer»

(i) 43.123456789

Since 43.123456789 has terminating decimal expansion. Hence, its denominator is of the form 2^m x 5ⁿ, where m, n are non-negative integers.

(ii) \(43.\overline{123456789}\)

Since the given rational has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.

(iii) \(27.\overline{142857}\)

Since the given rational number has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.

(iv) 0.120120012000120000….

Since 0.120120012000120000…. has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.