Prime Factorization of Denominators from Given Decimal Expansions

Video Explanation

Watch the video below for the complete explanation:

Solution

Question: What can you say about the prime factorizations of the denominators of the following rational numbers?

Important Facts

• A terminating decimal has denominator of the form 2^m × 5ⁿ.
• A non-terminating repeating decimal has denominator with primes other than 2 and 5.
• A non-terminating non-repeating decimal is irrational.

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(i) 43.123456789

This decimal is terminating because it ends after a finite number of digits.

∴ The denominator (in lowest form) has only prime factors 2 and/or 5.

Denominator is of the form 2^m × 5ⁿ.

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(ii) 43.123456789

This decimal is also terminating.

∴ Its denominator contains only the prime factors 2 and/or 5.

Denominator is of the form 2^m × 5ⁿ.

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(iii) 27.142857

The digits repeat in a fixed block (142857).

∴ This is a non-terminating repeating decimal.

So, the denominator contains prime factors other than 2 and 5.

Denominator is NOT of the form 2^m × 5ⁿ.

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(iv) 0.120120012000120000…

The decimal is non-terminating and non-repeating.

Such a decimal represents an irrational number.

∴ It cannot be expressed as a rational number.

Hence, no prime factorization of a denominator exists.

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Final Answer

(i) Denominator has only 2 and/or 5 as prime factors
(ii) Denominator has only 2 and/or 5 as prime factors
(iii) Denominator has primes other than 2 and 5
(iv) Not a rational number (no denominator exists)

Conclusion

Thus, the nature of the decimal expansion directly determines the prime factorization of the denominator of a rational number.