If p/q is in lowest form and q = 2^m × 5^n × 13^r, where r > 0, what type of decimal expansion will it have?
Answer and explanation
Correct answer: Non-terminating recurring
The relevant theorem states that a rational number p/q in lowest terms has a terminating decimal if and only if the reduced denominator q is of the form 2^a × 5^b. Here q also contains 13^r, and r is positive, so a factor 13 remains in the denominator. Lowest form guarantees that this factor cannot be cancelled by the numerator. Therefore q cannot be transformed into a power of 10, and the decimal expansion does not end. Since p/q is rational, its non-terminating decimal must eventually repeat; it cannot be non-terminating non-recurring. Thus option B is correct. Options A and D would require the absence of the factor 13, while option C does not describe the decimal behaviour of a rational number.
Frequently asked questions
What is the correct answer to this question?
Non-terminating recurring
Why is this the correct answer?
The relevant theorem states that a rational number p/q in lowest terms has a terminating decimal if and only if the reduced denominator q is of the form 2^a × 5^b. Here q also contains 13^r, and r is positive, so a factor 13 remains in the denominator. Lowest form guarantees that this factor cannot be cancelled by the numerator. Therefore q cannot be transformed into a power of 10, and the decimal expansion does not end. Since p/q is rational, its non-terminating decimal must eventually repeat; it cannot be non-terminating non-recurring. Thus option B is correct. Options A and D would require the absence of the factor 13, while option C does not describe the decimal behaviour of a rational number.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Real Numbers. Topic: Decimal expansion of rational numbers.
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