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goodyear2013
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If p, q, and r are positive integers such that q is a factor Updated on: 06 Jul 2014, 11:17
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.
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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

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E
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If p, q, and r are positive integers such that q is a factor 06 Jul 2014, 11:36
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goodyear2013
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

Show Spoiler
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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If p, q, and r are positive integers such that q is a factor 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 20 Dec 2014, 10:22
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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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If p, q, and r are positive integers such that q is a factor 20 Dec 2014, 10:35
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veerdonjuan
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 20 Dec 2014, 10:44
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: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
veerdonjuan
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 15 Jun 2017, 03:15
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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

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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If p, q, and r are positive integers such that q is a factor 17 Jun 2017, 05:25
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goodyear2013
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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If p, q, and r are positive integers such that q is a factor 24 Aug 2019, 07:44
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goodyear2013
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

Show Spoiler
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.
How did you go about choosing these numbers ? At the time of examination how to decide which numbers to choose Bunuel
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If p, q, and r are positive integers such that q is a factor 26 Aug 2019, 01:42
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Bunuel
goodyear2013
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

Show Spoiler
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.
How did you go about choosing these numbers ? At the time of examination how to decide which numbers to choose Bunuel

Good question. We have number of articles on this. Check below.

Number plugging:


How to Do Math on the GMAT Without Actually Doing Math
The Power of Estimation for GMAT Quant
How to Plug in Numbers on GMAT Math Questions
Number Sense for the GMAT
Can You Use a Calculator on the GMAT?
Why Approximate?
GMAT Math Strategies — Estimation, Rounding and other Shortcuts
The 4 Math Strategies Everyone Must Master, Part 1 (1. Test Cases and 2. Choose Smart Numbers.)
The 4 Math Strategies Everyone Must Master, part 2 (3. Work Backwards and 4. Estimate)
Intelligent Guessing on GMAT
How to Avoid Tedious Calculations on the Quantitative Section of the GMAT
GMAT Tip of the Week: No Calculator? No Problem.
The Importance of Sorting Answer Choices on the GMAT

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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If p, q, and r are positive integers such that q is a factor 26 Aug 2019, 01:55
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goodyear2013
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

Show Spoiler
Premier P716

Given:
1. p, q, and r are positive integers
2. q is a factor of r
3. r is a multiple of p

Asked: Which of the following must be an integer?

r = k1*q
r = k2*p
r = k1*q = k2*p where k1 & k2 are integers

Let us take r = 12; p=4; q=3
A. (p + q)/r
7/12
Not necessarily an integer
B. (r + p)/q
16/3
Not necessarily an integer
C. p/q
4/3
Not necessarily an integer
D. pq/r
1
Let us take r = 12 ; p=2; q =3
6/12
Not necessarily an integer
E. r(p + q)/pq
12*5/6=10
r (1/p + 1/q) = r/p + r/q = k1 + k2 = integer

IMO E
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If p, q, and r are positive integers such that q is a factor 24 Mar 2020, 10:16
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The first thing is about writing the information in the right way. We know that:
r= N*q
r= M*p 
In whihc M and N are just random numbers. 
At this point to make the problem easier we are going to take the simplest number made out of 4 different prime factor: 210 ( 2*3*5*7 ) and put p=2 and q=3. 
At this point we go to test he cases. 

E)210*5 / 6 = 175
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If p, q, and r are positive integers such that q is a factor 03 Feb 2025, 14:57
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Hi Bunel, maybe I think incorrectly about it by why isn't the answer C?

We know:
1) r=aq (where a is a factor, i.e. ingeter)
2) p=br (where b is a factor, i.e. integer)

So: p=abq
p/q = ab = integer * integer = integer

Could you kindly clarify what you thinking is / where my reasoning is incorrect?

Thank you in advance!

Bunuel
goodyear2013
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

Show Spoiler
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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GMAT4goodunis
Hi Bunel, maybe I think incorrectly about it by why isn't the answer C?

We know:
1) r=aq (where a is a factor, i.e. ingeter)
2) p=br (where b is a factor, i.e. integer)

So: p=abq
p/q = ab = integer * integer = integer

Could you kindly clarify what you thinking is / where my reasoning is incorrect?

Thank you in advance!

Bunuel
goodyear2013
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

Show Spoiler
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.
r is a multiple of p means r = p * integer, not p = r * integer.
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