The decimal expansion of will be terminating precisely when the prime factors of are only 2, only 5 or both 2 and 5. Can you explain why?

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KnowledgeBoat · Accepted Answer

The decimal expansion of terminates if and only if the denominator q (in lowest terms) has only 2 and/or 5 as its prime factors. Reason : Our number system is base-10, and 10 = 2 × 5. So, any denominator that is a power of 10 produces a terminating decimal. If q has only the prime factors 2 and/or 5, we can multiply both the numerator and the denominator by a suitable number to make the denominator a power of 10. This gives a terminating decimal. Example 1 : ⇒ (terminating). Example 2 : ⇒ (terminating). But, if q has any prime factor other than 2 or 5 (such as 3, 7, 11, etc.), no multiplication can make the denominator a power of 10. So, the long division will never give a remainder of 0, and the decimal expansion will be non-terminating. Example 3 : ⇒ (non-terminating, repeating). Here 3 is a prime factor other than 2 or 5. Hence, the decimal expansion of terminates only when the denominator q (in lowest form) has 2 and/or 5 as its only prime factors, because only then can it be expressed as a fraction with a power of 10 in the denominator.

Mathematics

The decimal expansion of will be terminating precisely when the prime factors of are only 2, only 5 or both 2 and 5. Can you explain why?

Whole Numbers

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