If \(r\) and \(s\) are positive integers, can the fraction \(\dfrac{r}{s}\) be expressed as a decimal with only a finite

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If \(r\) and \(s\) are positive integers, can the fraction \(\dfrac{r}{s}\) be expressed as a decimal with only a finite number of nonzero digits?

(1) \(s\) is a factor of \(100.\)
(2) \(r\) is a factor of \(100.\)

Answer: A

Source: Official Guide

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Gmat_mission wrote:
Sun Sep 19, 2021 12:41 pm
If \(r\) and \(s\) are positive integers, can the fraction \(\dfrac{r}{s}\) be expressed as a decimal with only a finite number of nonzero digits?

(1) \(s\) is a factor of \(100.\)
(2) \(r\) is a factor of \(100.\)

Answer: A

Source: Official Guide
Target question: Is r/s a terminating decimal?

Statement 1: s is a factor of 100
There's a nice rule that says something like,
If the prime factorization of the denominator contains only 2's and/or 5's, then the decimal version of the fraction will be a terminating decimal.
Since 100 = (2)(2)(5)(5), any factor of 100 will contain only 2's and/or 5's (or the denominator can be 1, in which case the decimal will definitely terminate).
Since the denominator of r/s must contain only 2's and/or 5's, r/s must be a terminating decimal
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: r is a factor of 100
There are several pairs of values that meet this condition. Here are two:
Case a: r = 1 and s = 4, in which case r/s = 1/4 = 0.25, which is a terminating decimal
Case b: r = 1 and s = 3, in which case r/s = 1/3 = 0.333.., which is not a terminating decimal
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

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Brent
Brent Hanneson - Creator of GMATPrepNow.com
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