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

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Senior Manager
Joined: 21 Oct 2013
Posts: 419
If p, q, and r are positive integers such that q is a factor  [#permalink]

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Updated on: 06 Jul 2014, 10:17
3
10
00:00

Difficulty:

65% (hard)

Question Stats:

62% (02:26) correct 38% (02:00) wrong based on 226 sessions

### HideShow timer Statistics

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, 08:58.
Last edited by Bunuel on 06 Jul 2014, 10:17, edited 1 time in total.
Edited the question.
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Posts: 50712
Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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06 Jul 2014, 10:36
3
1
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. $$\frac{r(p + q)}{pq}= \frac{rp+rq}{pq}=\frac{rp}{pq}+\frac{rq}{pq}=\frac{r}{q}+\frac{r}{p}=integer+integer=integer.$$

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

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08 Jul 2014, 20:40
2
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

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

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20 Dec 2014, 09:22
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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Posts: 50712
Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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20 Dec 2014, 09:35
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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20 Dec 2014, 09:44
Thankyou Bunuel !!I think I need good sleep
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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Joined: 13 Aug 2014
Posts: 4
If p, q, and r are positive integers such that q is a factor  [#permalink]

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15 Jun 2017, 01:53
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
Math Expert
Joined: 02 Sep 2009
Posts: 50712
Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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15 Jun 2017, 02:08
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

Merging topics. Please refer to the discussion above.
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Re: If p, q, and r are positive integers such that q is a factor  [#permalink]

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15 Jun 2017, 02:15
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'.

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

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17 Jun 2017, 04:25
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.

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

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12 Sep 2018, 19:33
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Re: If p, q, and r are positive integers such that q is a factor &nbs [#permalink] 12 Sep 2018, 19:33
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