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# If r and s are positive integers, can the fraction r/s be expressed as

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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If r and s are positive integers, can the fraction r/s be expressed as

If r and s are positive integers, can the fraction 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

This is a commonly tested type of question.
If we modify the question, we get a decimal with only a finite number of nonzero digits=terminating decimal. So, it is referring to things such as 0.2=1/5, 0.5=1/2, 0.21=21/(2^2)(5^2). But in order to become a terminating decimal, the denominator has to have 2 or 5 as their prime factors; as the question asks whether the prime factor of s is composed of only 2 or 5 in r/s, we only need to know the value of s.
In condition 1, 100=(2^2)(5^2). The prime factors only include 2 or 5, so the condition is sufficient and the answer becomes (A).

Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
The question asks if r/s can be expressed as a decimal with only a finite number of nonzero digits. So, it's asking if r/s can be expressed with a finite number of nonzero digits and nothing else (therefore, there can be no zeroes in the decimal). So, 10, for example, would not work (there's a zero in 10).

Or are we supposed to read the "only" to refer only to the "finite number" part of the question?
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
momonmoprob wrote:
The question asks if r/s can be expressed as a decimal with only a finite number of nonzero digits. So, it's asking if r/s can be expressed with a finite number of nonzero digits and nothing else (therefore, there can be no zeroes in the decimal). So, 10, for example, would not work (there's a zero in 10).

Or are we supposed to read the "only" to refer only to the "finite number" part of the question?

"Only" refers to finite, only.
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If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
Bunuel wrote:
momonmoprob wrote:
The question asks if r/s can be expressed as a decimal with only a finite number of nonzero digits. So, it's asking if r/s can be expressed with a finite number of nonzero digits and nothing else (therefore, there can be no zeroes in the decimal). So, 10, for example, would not work (there's a zero in 10).

Or are we supposed to read the "only" to refer only to the "finite number" part of the question?

"Only" refers to finite, only.

I find that a bit confusing, is there some way I can avoid this confusion in the future? To me the wording is pretty clear, the decimal should only have a finite number of nonzero digits, in which case there should be a finite number of nonzero digits and nothing else (no zeroes).

How are we supposed to know during the test what they mean?

EDIT: Further, if they meant for the "only" to refer to "finite," only, then they simply should have wrote the question as "...can r/s can be expressed as a decimal with a finite number of nonzero digits?" That would have gotten the intended message across.
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
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momonmoprob wrote:
Bunuel wrote:
momonmoprob wrote:
The question asks if r/s can be expressed as a decimal with only a finite number of nonzero digits. So, it's asking if r/s can be expressed with a finite number of nonzero digits and nothing else (therefore, there can be no zeroes in the decimal). So, 10, for example, would not work (there's a zero in 10).

Or are we supposed to read the "only" to refer only to the "finite number" part of the question?

"Only" refers to finite, only.

I find that a bit confusing, is there some way I can avoid this confusion in the future? To me the wording is pretty clear, the decimal should only have a finite number of nonzero digits, in which case there should be a finite number of nonzero digits and nothing else (no zeroes).

How are we supposed to know during the test what they mean?

EDIT: Further, if they meant for the "only" to refer to "finite," only, then they simply should have wrote the question as "...can r/s can be expressed as a decimal with a finite number of nonzero digits?" That would have gotten the intended message across.

Hello momonmoprob

The phrases "with only a finite number of nonzero digits" and "with a finite number of nonzero digits" are equal.

"To me the wording is pretty clear, the decimal should only have a finite number of nonzero digits"
this is not correct because the word "only" can not modify words and do not stand next to them:

"with only a finite number of nonzero digits" - here word "only" modifies "finite number"
"with a finite number of only nonzero digits" - here word "only" modifies "nonzero digits"
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
What if r=25 and s=25

Then 25/25=1

It has no decimal point in this case.

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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
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Goindanij wrote:
What if r=25 and s=25

Then 25/25=1

It has no decimal point in this case.

Posted from my mobile device

An integer IS a decimal with a finite number of nonzero digits. For example, integer 1 has 1 (finite) number of digits.
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
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asveaass wrote:
If r and s are positive integers, can the fraction 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

Solution:

Question Stem Analysis:

We need to determine whether r/s can be expressed as a decimal with only a finite number of nonzero digits, i.e., whether r/s can be expressed as a terminating decimal. Recall that if a fraction in lowest terms can be expressed as a terminating decimal, the denominator of the fraction must have prime factors of only 5 and/or 2..

Statement One Alone:

Since s is a factor of 100 = 2^2 x 5^2, then s itself has prime factors of only 2 and/or 5. Since s is the denominator of the fraction r/s, we see that r/s can be indeed expressed as a terminating decimal.

Statement Two Alone:

Even though we know r is a factor of 100, we don’t know anything about s, the denominator of the fraction r/s. We can’t determine whether r/s can be expressed as a terminating decimal since it relies on s, the denominator of the faction, rather than r, the numerator of the fraction.

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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
asveaass wrote:
If r and s are positive integers, can the fraction 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

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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
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If r and s are positive integers, can the fraction r/s be expressed as a decimal with only a finite number of nonzero digits?

We can re frame the question as is r/s a terminating decimal or not?

if the only prime factors of s are either 2 or 5 or both 2 and 5, then we can conclude that r/s will terminating.

For example: 1/5, 11/32, 7/25, 17/50 are all terminating decimals. When we prime factorize the denominators we see that the only prime factors are either 2 or 5 or both.

If s has prime factors other than 2 and 5 , then we need to check if its still there as the prime factor of denominator of the reduced/simplified form of r/s. If yes, then its a non terminating decimal.

For example: 3/14, 5/22, 13/70 are non terminating decimal.

3/14 = 3/2*7 is non terminating as 7 is a prime factor of the denominator.
5/22= 5/ 2*11 is a non terminating decimal as 11 is a prime factor of the denominator.

lets try an example 14/35 .
14/35 =14/(5*7)
Since denominator have a prime factor of 7, we need to cross check it's there in the simplified form as well.
Simplified fraction is 2/5 =.4 which is terminating decimal as 7 get cancelled in the numerator and denominator. That is the reason why we need to confirm by factorizing the denominator of simplified fraction and see if any prime factor other than 2 or 5 is there or not.

(1) s is a factor of 100

Since the prime factors of 100 are only 2 and 5 . Therefore we can conclude that prime factors of s also will be either 2 or 5 or both. That means r/s will be terminating decimal.
Hence statement 1 alone is sufficient.

(2) r is a factor of 100
Here we don't have any info about the denominator s. So we cant decide whether r/s is terminating or not. Hence Statement 2 alone is insufficient.

Option A is the correct answer.

Thanks,
Clifin J Francis,
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
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asveaass wrote:
If r and s are positive integers, can the fraction 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

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

Cheers,
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
halloanupam wrote:
But it has been said the 'decimal with finite number of non-zero digits'. Now if we take 2/50 then it will be 0.04, which means its a finite decimal but definitely it does not have all the non-zero digits after decimal point. So, can anybody explain?

I was thinking the same with you, but I suddenly got that what it means is , the number of nonzenro digits should be finite, not all of the digits should be nonzero. Now you Got it?

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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
I kept thinking "what if R or S" were pi? Since its an irrational number it cannot be expressed as a finite decimal or fraction, so what is R = pi, then even if S was a factor of 100, pie times any one of these factors would still be irrational, right?
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
elizabethsmith719 wrote:
I kept thinking "what if R or S" were pi? Since its an irrational number it cannot be expressed as a finite decimal or fraction, so what is R = pi, then even if S was a factor of 100, pie times any one of these factors would still be irrational, right?

You should read the stem carefully: If r and s are positive integers...
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
Bunuel wrote:
asveaass wrote:
If r and s are positive integers, can the fraction 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

I don't understand the answer explanation in the OG, could someone please explain in detail?

THEORY:

Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.

For example $$\frac{x}{2^n5^m}$$, (where x, n and m are integers) will always be terminating decimal.

(We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction $$\frac{6}{15}$$ has 3 as prime in denominator and we need to know if it can be reduced.)

Questions testing this concept:
https://gmatclub.com/forum/does-the-dec ... 89566.html
https://gmatclub.com/forum/any-decimal- ... 01964.html
https://gmatclub.com/forum/if-a-b-c-d-a ... 25789.html
https://gmatclub.com/forum/700-question-94641.html
https://gmatclub.com/forum/is-r-s2-is-a ... 91360.html
https://gmatclub.com/forum/pl-explain-89566.html
https://gmatclub.com/forum/which-of-the ... 88937.html

BACK TO THE ORIGINAL QUESTION:
If r and s are positive integers, can the fraction r/s be expressed as a decimal with only a finite number of nonzero digits?

(1) s is a factor of 100. Factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50 and 100. All these numbers are of the form $$2^n5^m$$ (for example 1=2^0*5^0, 2=2^1*5^0, ...), therefore no matter what is the value of r, r/s will always will be terminating decimal. Sufficient.

(2) r is a factor of 100. We need to know about the denominator. Not sufficient.

Hope it's clear.

How can the number be a terminating decimal with a finite # of nonzero digits if the denominator has 1? It will become an integer in that case. Can someone please help?
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
sharmashagun770 wrote:
Bunuel wrote:
asveaass wrote:
If r and s are positive integers, can the fraction 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

I don't understand the answer explanation in the OG, could someone please explain in detail?

THEORY:

Reduced fraction $$\frac{a}{b}$$ (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only $$b$$ (denominator) is of the form $$2^n5^m$$, where $$m$$ and $$n$$ are non-negative integers. For example: $$\frac{7}{250}$$ is a terminating decimal $$0.028$$, as $$250$$ (denominator) equals to $$2*5^3$$. Fraction $$\frac{3}{30}$$ is also a terminating decimal, as $$\frac{3}{30}=\frac{1}{10}$$ and denominator $$10=2*5$$.

Note that if denominator already has only 2-s and/or 5-s then it doesn't matter whether the fraction is reduced or not.

For example $$\frac{x}{2^n5^m}$$, (where x, n and m are integers) will always be terminating decimal.

(We need reducing in case when we have the prime in denominator other then 2 or 5 to see whether it could be reduced. For example fraction $$\frac{6}{15}$$ has 3 as prime in denominator and we need to know if it can be reduced.)

Questions testing this concept:
https://gmatclub.com/forum/does-the-dec ... 89566.html
https://gmatclub.com/forum/any-decimal- ... 01964.html
https://gmatclub.com/forum/if-a-b-c-d-a ... 25789.html
https://gmatclub.com/forum/700-question-94641.html
https://gmatclub.com/forum/is-r-s2-is-a ... 91360.html
https://gmatclub.com/forum/pl-explain-89566.html
https://gmatclub.com/forum/which-of-the ... 88937.html

BACK TO THE ORIGINAL QUESTION:
If r and s are positive integers, can the fraction r/s be expressed as a decimal with only a finite number of nonzero digits?

(1) s is a factor of 100. Factors of 100 are: 1, 2, 4, 5, 10, 20, 25, 50 and 100. All these numbers are of the form $$2^n5^m$$ (for example 1=2^0*5^0, 2=2^1*5^0, ...), therefore no matter what is the value of r, r/s will always will be terminating decimal. Sufficient.

(2) r is a factor of 100. We need to know about the denominator. Not sufficient.

Hope it's clear.

How can the number be a terminating decimal with a finite # of nonzero digits if the denominator has 1? It will become an integer in that case. Can someone please help?

I think your doubt is already addressed in this topic: an integer IS a decimal with a finite number of nonzero digits. For example, integer 51 has 2 (finite) number of digits.
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Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
Bunuel is an Integer considered a terminating decimal on GMAT? If r and s are not in fraction form but reduce to an integer, will the answer still be A?
Re: If r and s are positive integers, can the fraction r/s be expressed as [#permalink]
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