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# If r and s are positive integers, is r/s an integer?

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Manager
Joined: 02 Nov 2008
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If r and s are positive integers, is r/s an integer?  [#permalink]

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10 Nov 2009, 14:24
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If A and B are positive integers, is a/b an integer?

(1) Every factor of B is also a factor of A
(2) Every prime factor of B is also a prime factor of A
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Joined: 02 Sep 2009
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10 Nov 2009, 14:44
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chicagocubsrule wrote:
If A and B are positive integers, is a/b an integer?
1) Every factor of B is also a factor of A
2) Every prime factor of B is also a prime factor of A

(1) If every factor of B is also factor of A, then in fraction A/B, B will just be reduced and we get an integer. Sufficient.

(2) The powers of primes factors of B could be higher than powers of prime factors of A. eg 25/125=1/5 not an integer. Not sufficient.

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10 Nov 2009, 21:38
1
Bunuel wrote:
chicagocubsrule wrote:
If A and B are positive integers, is a/b an integer?
1) Every factor of B is also a factor of A
2) Every prime factor of B is also a prime factor of A

(1) If every factor of B is also factor of A, then in fraction A/B, B will just be reduced and we get an integer. Sufficient.

(2) The powers of primes factors of B could be higher than powers of prime factors of A. eg 25/125=1/5 not an integer. Not sufficient.

+1
so even though they share the same prime factors doesn't mean the have the same powers.
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Joined: 08 Nov 2009
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11 Nov 2009, 19:07
Bunuel,

looking at (2) I would say that with the current frasing this statement is sufficient. I would agree with your answer if (2) said "...every distinct prime factor..."

When the GMAT refers to prime factors, should I assume that it is referring only to distinct ones (eg 8 has three prime factors, 2, 2, and 2, but only one distinct prime factor)?
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12 Nov 2009, 05:07
Marco83 wrote:
Bunuel,

looking at (2) I would say that with the current frasing this statement is sufficient. I would agree with your answer if (2) said "...every distinct prime factor..."

When the GMAT refers to prime factors, should I assume that it is referring only to distinct ones (eg 8 has three prime factors, 2, 2, and 2, but only one distinct prime factor)?

Welcome to the forum Marco83.

For the question. (2) states: "Every prime factor of B is also a prime factor of A." Which exactly means distinct prime factors, how else it could be? In your example: we can say that every prime factor of 8 is a factor of 30, as 8 has only one prime, which is 2 and 2 IS a factor of 30.

Let's suppose prime factorization of B is: $$B=x^p*y^q*z^r$$ from the statement we know that A's prime factors must be x, y, and z. But we don't know the powers of x, y, z in A or B. So in the fraction $$\frac{A}{B}$$ B may or may not be reduced. If the power of either of prime is higher in B we'll get the fraction.

Also note that we are told that B's primes are A's primes too, but not vise versa, though this doesn't affect the conclusion here.

Hope it helps. Please tell if it needs further clarification.
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12 Nov 2009, 05:43
If B=x^p*y^q*z^r, the prime factors of B are
x
x
x
.
.
.
(p times)
y
y
y
.
.
(q times)
z
z
z
.
.
(r times)

While the distinct prime factors are simply x, y, and z.

If the second clue was frased as "...the DISTINCT prime factors...", I would agree with you that by having different exponents on each distinct prime factor A/B could result in a fraction, but since the clue is frased as "...the prime factors..." to me this means that A=x^p1*x^q1*z^r1*other primes, where p1>=p, q1>=q, and r1>=r. This leads to A/B=x^(p1-p)*y^(q1-q)*z^(r1-r)*other primes, which is indeed an integer.

What do you think?
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12 Nov 2009, 05:58
Marco83 wrote:
If B=x^p*y^q*z^r, the prime factors of B are
x
x
x
.
.
.
(p times)
y
y
y
.
.
(q times)
z
z
z
.
.
(r times)

While the distinct prime factors are simply x, y, and z.

If the second clue was frased as "...the DISTINCT prime factors...", I would agree with you that by having different exponents on each distinct prime factor A/B could result in a fraction, but since the clue is frased as "...the prime factors..." to me this means that A=x^p1*x^q1*z^r1*other primes, where p1>=p, q1>=q, and r1>=r. This leads to A/B=x^(p1-p)*y^(q1-q)*z^(r1-r)*other primes, which is indeed an integer.

What do you think?

When we have $$B=x^p*y^q*z^r$$ we can say that prime factors of B are x, y, and z, which is the same as distinct prime factors of B. When you say that primes are x, p times it still means that the prime is only x, as x in any power higher than 1 is not the prime any more.
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26 Aug 2010, 11:59
Hi Bunuel,

How the two statements, A and B are different? What I understood from the options are - only the values of the factors for B are same as A and not the quantity.

Could you please explain. I chose E. The way you interpreted for option B, I interpreted for option A as well. Hence found the data to be insufficient.

Thanks.
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26 Aug 2010, 12:23
amp0201 wrote:
Hi Bunuel,

How the two statements, A and B are different? What I understood from the options are - only the values of the factors for B are same as A and not the quantity.

Could you please explain. I chose E. The way you interpreted for option B, I interpreted for option A as well. Hence found the data to be insufficient.

Thanks.

Statement (1) says: every factor of B is also a factor of A, as B is a factor of B itself, then it means that B is a factor of A. So in fraction A/B, B will just be reduced and we get an integer.

Hope it's clear.
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29 Aug 2010, 05:54
thank you....very clear
Manager
Joined: 07 Feb 2010
Posts: 129
if r & s are positive integers  [#permalink]

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26 Dec 2010, 06:37
if r & s are positive integers is r/s an integer
a) every factor of s is also a factor of r
b) every prime factor of s is also a prime factor of r
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Joined: 12 Dec 2010
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Schools: Wharton, Columbia, Booth, NYU

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26 Dec 2010, 14:14
1
Question here is A / B = Int, given A, B = positive integers
=> For A / B to be an integer, A > B

Now lets look at choices
a. Every factor of B is also a factor of A
=> A >= B and A is divisble by B (Sufficient)

b. Every prime factor of B is also prime factor of A
=> A could be greater or less than B (e.g. A = 3, B = 9 etc)
Not Suff.

OA: A
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11 Jan 2011, 11:32
I chose for D.
Considering that both are sufficient but as you have mentioned the powers of prime factor in the denominator can be greater than numerator which results a fraction...
Got it...
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08 Jan 2012, 01:23

Why the answer is not D
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08 Jan 2012, 01:58
1
1) Each factor of S is also a factor of R ==> R is a multiple of S Sufficient

2) Every prime factor of S is also a prime factor of R ==> Insuff.
R & S might have prime factors : 2,3 ...
but when R/S Example: 4 * 9/81*4 is not an integer

Hence A
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08 Jan 2012, 02:05
gmatv11 wrote:
1) Each factor of S is also a factor of R ==> R is a multiple of S Sufficient

2) Every prime factor of S is also a prime factor of R ==> Insuff.
R & S might have prime factors : 2,3 ...
but when R/S Example: 4 * 9/81*4 is not an integer

Hence A

Thanks.....+1 Kudos for you
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08 Jan 2012, 18:02
1. Obvious rule for division. Suff.
2. If r = 2x3x7 and s=2x3x7, good. If r = 2x3x7 and s=2x3x7x7, not good. Insuff.

A.
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Re: If r and s are positive integers, is r/s an integer?  [#permalink]

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01 Jul 2018, 15:27
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Re: If r and s are positive integers, is r/s an integer? &nbs [#permalink] 01 Jul 2018, 15:27
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