If (p/q) is in simplest form and (q=5^8), what is the maximum number of decimal places in the terminating decimal?
Answer and explanation
Correct answer: (8)
A rational number in lowest terms has a terminating decimal only when the prime factors of its denominator are 2 and/or 5. Here q = 5^8. To express the denominator as a power of 10, multiply numerator and denominator by 2^8: 5^8 × 2^8 = 10^8. Thus the decimal can have at most eight places. It may have fewer places if the numerator causes cancellation after conversion, but eight is the maximum possible. Therefore option C is correct. Options A and B do not supply enough factors of 2 to form 10^8, while option D overestimates the required power. The conclusion follows directly from the terminating-decimal theorem.
Frequently asked questions
What is the correct answer to this question?
(8)
Why is this the correct answer?
A rational number in lowest terms has a terminating decimal only when the prime factors of its denominator are 2 and/or 5. Here q = 5^8. To express the denominator as a power of 10, multiply numerator and denominator by 2^8: 5^8 × 2^8 = 10^8. Thus the decimal can have at most eight places. It may have fewer places if the numerator causes cancellation after conversion, but eight is the maximum possible. Therefore option C is correct. Options A and B do not supply enough factors of 2 to form 10^8, while option D overestimates the required power. The conclusion follows directly from the terminating-decimal theorem.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Decimal representation.
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