Let x= $\frac{p}{q}$ be a rational number, such that the prime factorisation of q is not of the form $2^{m} 5^{n}$ , where n, m are non-negative integers. Then, x has a decimal expansion which is ___.

non-terminating repeating (recurring)

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terminating non-repeating (non-recurring)

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real integer

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none of these

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Solution

The correct option is A non-terminating repeating (recurring)
A rational $\frac{p}{q}$ has a decimal expansion which is terminating only when the denominator can be prime factorised in the form $2^{m} 5^{n}$ . If the denominator has any other prime factor, then the decimal becomes non-terminating and repeating.

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