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If p, q, and r are positive integers such that q is a factor

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If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post Updated on: 06 Jul 2014, 11:17
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If p, q, and r are positive integers such that q is a factor of r, and r is a multiple of p, which of the following must be an integer?

A. (p + q)/r
B. (r + p)/q
C. p/q
D. pq/r
E. r(p + q)/pq

Premier P716

Originally posted by goodyear2013 on 06 Jul 2014, 09:58.
Last edited by Bunuel on 06 Jul 2014, 11:17, edited 1 time in total.
Edited the question.
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Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post 06 Jul 2014, 11:36
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goodyear2013 wrote:
If p, q, and r are positive integers such that q is a factor of r, and r is a multiple of p, which of the following must be an integer?

A. (p + q)/r
B. (r + p)/q
C. p/q
D. pq/r
E. r(p + q)/pq

Premier P716


We have that r is a multiple of both q and p.

A. (p + q)/r. Not necessarily true. Consider r = 3, and p = q = 1.

B. (r + p)/q. Not necessarily true. Consider r = 6, and p = 2 and q=3.

C. p/q. Not necessarily true. Consider r = 6, and p = 2 and q=3.

D. pq/r. Not necessarily true. Consider r = 3, and p = q = 1.

E. \(\frac{r(p + q)}{pq}= \frac{rp+rq}{pq}=\frac{rp}{pq}+\frac{rq}{pq}=\frac{r}{q}+\frac{r}{p}=integer+integer=integer.\)

Answer: E.
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Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post 08 Jul 2014, 21:40
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Let p, q, r be 2, 4, 24

Cross check the conditions given

4 is a factor of 24 ? .......... YES

24 is a multiple of 2? .......... YES

Check the options:

A: \(\frac{2+4}{24}\) ......... Fail

B: \(\frac{24 + 2}{4}\) ......... Fail

C: \(\frac{4}{24}\) ............ Fail

D: \(\frac{2*4}{24}\) ........... Fail

E: \(\frac{24(2+4)}{4*6}\) ................ Success

Answer = E
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If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post 20 Dec 2014, 10:22
I am not able to get the right answer :?

1)If q is a factor of r, then r=aq (a any constant )
2)If r is a multiple of p,then p=br (b any constant)

Now checking options
1. (p+q)/r= (br+r/a)/r= b+1/a .................... It can b an Integer if a=1
2. (r+p)/q= (r+br)/(r/a)= a(1+b)............ Always an Integer
3. p/q =rb/(r/a)=b/a ................................ Can be an Integer if b is divisible by a
4. r(p+q)/pq= r(rb+r/a)/(rb*r/a)=(ab+1)/b ...............:(


I think I m doing some basic mistake ,please tell.
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Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post 20 Dec 2014, 10:35
veerdonjuan wrote:
I am not able to get the right answer :?

1)If q is a factor of r, then r=aq (a any constant )
2)If r is a multiple of p,then p=br (b any constant)

Now checking options
1. (p+q)/r= (br+r/a)/r= b+1/a .................... It can b an Integer if a=1
2. (r+p)/q= (r+br)/(r/a)= a(1+b)............ Always an Integer
3. p/q =rb/(r/a)=b/a ................................ Can be an Integer if b is divisible by a
4. r(p+q)/pq= r(rb+r/a)/(rb*r/a)=(ab+1)/b ...............:(


I think I m doing some basic mistake ,please tell.


r is a multiple of p means r = bp, not p = br.
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Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post 20 Dec 2014, 10:44
:shock: Thankyou Bunuel !!I think I need good sleep :P
Please tell substituting a value is good (less time consuming & accurate ) approach or assuming variables
Bunuel wrote:
veerdonjuan wrote:
I am not able to get the right answer :?

1)If q is a factor of r, then r=aq (a any constant )
2)If r is a multiple of p,then p=br (b any constant)

Now checking options
1. (p+q)/r= (br+r/a)/r= b+1/a .................... It can b an Integer if a=1
2. (r+p)/q= (r+br)/(r/a)= a(1+b)............ Always an Integer
3. p/q =rb/(r/a)=b/a ................................ Can be an Integer if b is divisible by a
4. r(p+q)/pq= r(rb+r/a)/(rb*r/a)=(ab+1)/b ...............:(


I think I m doing some basic mistake ,please tell.


r is a multiple of p means r = bp, not p = br.

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If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post 15 Jun 2017, 02:53
If p, q, and r are positive integers such that q is a factor of r, and r is a multiple of p, which of the following must be an integer?

A. (p+q)/r
B. (r+p)/q
C. p/q
D. pq/r
E. r(p+q)/pq
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Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post 15 Jun 2017, 03:08
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Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post 15 Jun 2017, 03:15
aarushi00 wrote:
If p, q, and r are positive integers such that q is a factor of r, and r is a multiple of p, which of the following must be an integer?

A. (p+q)/r
B. (r+p)/q
C. p/q
D. pq/r
E. r(p+q)/pq


One of the simple way to solve such kind of questions is to use values for different variables given.

Lets Assume
r = 8
q = 2
p= Either 1 or 4. Need to assume two value since relationship between p & q is not given

Let's plug these values
A. (p+q)/r = 3/8 = Not an Integer
B. (r+p)/q = 12/2 or 9/2 = Not an Integer in 2nd Scenario
C. p/q = 1/2 or 2/2 = Not an Integer in 1st Scenario
D. pq/r = 2/8 or 8/8 = Not an Integer in 1st Scenario
E. r(p+q)/pq = 8.3/3 or 8.6/8 = Integer in Both the Scenarios'.

Hence Answer E
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Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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New post 17 Jun 2017, 05:25
goodyear2013 wrote:
If p, q, and r are positive integers such that q is a factor of r, and r is a multiple of p, which of the following must be an integer?

A. (p + q)/r
B. (r + p)/q
C. p/q
D. pq/r
E. r(p + q)/pq


Since q is a factor of r and r is a multiple of p:

r/q = integer; r/p = integer.

After scanning the choices, we see that we can simplify answer choice E, so let’s do that first:

r(p + q)/pq = (rp + rq)/pq = rp/pq + rq/pq = r/q + r/p

Since we’ve mentioned that r/q = integer and r/p = integer, r/q + r/p must equal an integer also. Thus, answer choice E is correct.

Answer: E
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Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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Re: If p, q, and r are positive integers such that q is a factor   [#permalink] 12 Sep 2018, 20:33
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