Question

If x is a number whose simplest form is \(\frac {p}{q}\), where p and q are integers and q ≠ 0, then x is a non-terminating repeating decimal only when q is not of the form ________

(a) 2^m×2^n

(b) 5^m×5^n

(c) 2^m×5^n

(d) 3^m×4^n

I had been asked this question in an international level competition.

My question is taken from Irrational and Rational Numbers in division Real Numbers of Mathematics – Class 10

Answer 1

Correct answer is (c) 2^m×5^n

To explain: Let’s, take a number where q is of the form 2^m×5^n, say 2^50×5^10and p can be any integer

\(\frac {p}{2^{50}\times 5^{10}} = \frac {p \times 5^{40}}{10^{50}}\)

The number \(\frac {p \times 5^{40}}{10^{50}}\)  will terminate after 50 decimal places.

Hence, if q is of the form 2^m×5^n, it will terminate after some decimal places.