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If r and s are positive integers, is r/s a terminating decimal?

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If r and s are positive integers, is r/s a terminating decimal?
1) r is a factor of 100
2) s is a factor of 500
[Reveal] Spoiler: OA

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If r and s are positive integers, is r/s a terminating decimal? [#permalink]

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New post 04 Oct 2016, 07:16
MathRevolution wrote:
If r and s are positive integers, is r/s a terminating decimal?
1) r is a factor of 100
2) s is a factor of 500


To determine whether r/s is a terminating decimal, we need to know what do we have at the denominator.

if 3,7, etc. Then answer is NO and if 2,5,4, etc, the answer is YES.

Only statement 2 tells us that s is a factor of 500. hence, we will get a definite answer YES. Sufficient.
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Re: If r and s are positive integers, is r/s a terminating decimal? [#permalink]

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New post 04 Oct 2016, 07:20
I think it is a definite YES for statement 2.

s can have prime factors of either 2, 5 or both.

Always yields terminating decimal.


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Re: If r and s are positive integers, is r/s a terminating decimal? [#permalink]

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MathRevolution wrote:
If r and s are positive integers, is r/s a terminating decimal?

1) r is a factor of 100
2) s is a factor of 500


Target question: Is r/s a terminating decimal?

Statement 1: 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 1 is NOT SUFFICIENT

Statement 2: s is a factor of 500
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 500 = (2)(2)(5)(5)(5), any factor of 500 will contain only 2's and/or 5'2 (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 2 is SUFFICIENT

Answer:
[Reveal] Spoiler:
B


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Re: If r and s are positive integers, is r/s a terminating decimal? [#permalink]

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New post 06 Oct 2016, 05:39
==> If you change the original condition and the problem, in order for r/s to be the terminating decimal, prime factors of s should be 2 or 5. In the case of 2), from 500=2^25^3, prime factors of 500 should always be either 2 or 5, hence yes, and sufficient. The answer is B.
Answer: B
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Re: If r and s are positive integers, is r/s a terminating decimal? [#permalink]

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New post 12 Dec 2017, 12:33
MathRevolution wrote:
If r and s are positive integers, is r/s a terminating decimal?
1) r is a factor of 100
2) s is a factor of 500


When solving this problem, we should remember that there is a special property about fractions that allows their decimal equivalents to terminate. This property states:

In its most-reduced form, any fraction with a denominator whose prime factorization contains only 2s, only 5s, or both 2s and 5s, produces decimals that terminate. A denominator with any other prime factors produces decimals that do not terminate.

We must determine whether r/s is a terminating decimal, or in other words, whether s has only 2s, 5s, or both as prime factors.

Statement One Alone:

r is a factor of 100.

Since we do not have any information about s, statement one alone is not sufficient to answer the question.

Statement Two Alone:

s is a factor of 500.

Since 500 = 2^2 x 5^3 and s is a factor of 500, s will contain only 2s, 5s, or both as prime factors. If r/s is already in its most-reduced form, then r/s is a terminating decimal. If r/s is not in its most-reduced form, then the most-reduced form of r/s, say r’/s’, will also be a terminating decimal since s’ will then be a factor of s and it will contains only 2s, 5s or both as prime factors. Statement two alone is sufficient to answer the question.

Answer: B
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Re: If r and s are positive integers, is r/s a terminating decimal?   [#permalink] 12 Dec 2017, 12:33
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