Last visit was: 19 Nov 2025, 12:13 It is currently 19 Nov 2025, 12:13
Home
GMAT
Messages
Tests
Schools
Events
Rewards
Chat
My Profile
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
User avatar
yrozenblum
User avatar
Joined: 29 Mar 2020
Last visit: 20 Feb 2025
Posts: 27
Own Kudos:
1,725
 [104]
Given Kudos: 25
Posts: 27
Kudos: 1,725
 [104]
If p and q are different prime numbers, m and n are integers, and pq 09 Oct 2023, 03:24
6
Kudos
Add Kudos
98
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

65% (hard)

Question Stats:

66% (02:08) correct 34% (02:25) wrong based on 1054 sessions

History

Date
Time
Result
Not Attempted Yet
If p and q are different prime numbers, m and n are integers, and pq is a divisor of mp + nq, which of the following must be true?

I. p is a divisor of n.
II. pq is a divisor of mp.
III. p^2 is a divisor of mn.

A. I only
B. II only
C. III only
D. I and II
E. I and III

Show Spoiler
Attachment:
2024-01-27_14-10-56.png
2024-01-27_14-10-56.png [ 43.67 KiB | Viewed 15567 times ]
ShowHide Answer
Official Answer
D
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 Nov 2025
Posts: 105,390
Own Kudos:
778,336
 [18]
Given Kudos: 99,977
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 105,390
Kudos: 778,336
 [18]
If p and q are different prime numbers, m and n are integers, and pq 27 Jan 2024, 03:58
12
Kudos
Add Kudos
6
Bookmarks
Bookmark this Post
guddo
If p and q are different prime numbers, m and n are integers, and pq is a divisor of mp + nq, which of the following must be true?

I. p is a divisor of n.
II. pq is a divisor of mp.
III. p^2 is a divisor of mn.

A. I only
B. II only
C. III only
D. I and II
E. I and III

Show Spoiler
Attachment:
2024-01-27_14-10-56.png
Show more

Given that pq is a divisor of mp + nq implies:

    \(\frac{mp + nq}{pq} = integer\)

Splitting the fraction results in:

    \(\frac{mp}{pq} + \frac{nq}{pq} = integer\)

    \(\frac{m}{q} + \frac{n}{p} = integer\)

Since p and q are different prime numbers, we cannot have a case like 3/7 + 11/7 = 2. Therefore, for the sum to be an integer, both \(\frac{m}{q}\) and \(\frac{n}{p}\) must be integers. This means that q is a factor of m and p is a factor of n.

Let's evaluate the options:

I. p is a divisor of n.

As discussed above, this must be true.

II. pq is a divisor of mp.

The above says that \(\frac{mp}{pq}=\frac{m}{q}=integer\), and as discussed above, this must be true.

III. p^2 is a divisor of mn.

However, this is not necessarily true. For example, consider m = 2, n = 3, p = 3, and q = 2.

Answer: D.
User avatar
nisen20
User avatar
Joined: 16 Jun 2020
Last visit: 17 Nov 2025
Posts: 94
Own Kudos:
340
 [8]
Given Kudos: 504
Posts: 94
Kudos: 340
 [8]
If p and q are different prime numbers, m and n are integers, and pq 19 Apr 2024, 00:48
7
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Sonia2023
why can't m/q and n/p be fractions and their total is an integer?
Show more
­
\(\frac{mp + nq }{ pq} = k\) in which \(k\) is an integer.­

\(mp + nq = kpq\)

\(mp = kpq - nq\)

\(m = kq - \frac{nq}{p}\) in which \(m\) and \(kq\) are integers, so ­\(\frac{nq}{p}\) must be an integer.

as long as ­\(\frac{nq}{p}\) is an integer, ­\(\frac{n}{p}\) must be an integer because \(p\) and \(q\) are coprime.­
General Discussion
User avatar
gmatophobia
User avatar
Quant Chat Moderator
Joined: 22 Dec 2016
Last visit: 19 Nov 2025
Posts: 3,170
Own Kudos:
10,419
 [7]
Given Kudos: 1,861
Location: India
Concentration: Strategy, Leadership
Posts: 3,170
Kudos: 10,419
 [7]
If p and q are different prime numbers, m and n are integers, and pq 09 Oct 2023, 23:38
6
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
yrozenblum
If p and q are different prime numbers, m and n are integers, and pq is a divisor of mp + nq, which of the following must be true?

I. p is a divisor of n.

II. pq is a divisor of mp.

III. p² is a divisor of mn.

A) I only
B) II only
C) III only
D) I and II
E) I and III
Show more

pq is a divisor of mp + nq

\(\frac{mp + nq }{ pq}\) = integer

\(\frac{mp }{ pq} + \frac{nq }{ pq}\) = integer

\(\frac{m }{ q} + \frac{n }{ p}\) = integer

As q and p are different prime numbers, the above statement is possible when \(\frac{m }{ q}\) and \(\frac{n }{ p}\) are integers.

I. p is a divisor of n.

Inference: \(\frac{n}{p}\) = integer

This is correct. For the relationship mentioned in the question to hold true, \(\frac{n}{p}\) must be an integer. We can eliminate B and C.

II. pq is a divisor of mp.

Inference: \(\frac{mp}{pq} = \frac{m}{q} \) = integer

This is correct. For the relationship mentioned in the question to hold true, \(\frac{m}{q}\) must be an integer.

We don't have to check for III as none of the options have all three correct.

Option D
User avatar
Sonia2023
User avatar
Joined: 20 Feb 2022
Last visit: 12 Nov 2024
Posts: 61
Own Kudos:
27
 [1]
Given Kudos: 88
Location: India
Concentration: Finance, Other
Posts: 61
Kudos: 27
 [1]
If p and q are different prime numbers, m and n are integers, and pq 09 Nov 2023, 11:41
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
gmatophobia
yrozenblum
If p and q are different prime numbers, m and n are integers, and pq is a divisor of mp + nq, which of the following must be true?

I. p is a divisor of n.

II. pq is a divisor of mp.

III. p² is a divisor of mn.

A) I only
B) II only
C) III only
D) I and II
E) I and III

pq is a divisor of mp + nq

\(\frac{mp + nq }{ pq}\) = integer

\(\frac{mp }{ pq} + \frac{nq }{ pq}\) = integer

\(\frac{m }{ q} + \frac{n }{ p}\) = integer

As q and p are different prime numbers, the above statement is possible when \(\frac{m }{ q}\) and \(\frac{n }{ p}\) are integers.

I. p is a divisor of n.

Inference: \(\frac{n}{p}\) = integer

This is correct. For the relationship mentioned in the question to hold true, \(\frac{n}{p}\) must be an integer. We can eliminate B and C.

II. pq is a divisor of mp.

Inference: \(\frac{mp}{pq} = \frac{m}{q} \) = integer

This is correct. For the relationship mentioned in the question to hold true, \(\frac{m}{q}\) must be an integer.

We don't have to check for III as none of the options have all three correct.

Option D
Show more

gmatophobia - can you explain the highlighted part in bit more detail specifically addressing why can't m/q and n/p be fractions and their total is an integer?

Thanks a lot
User avatar
siddysir
User avatar
Joined: 29 May 2018
Last visit: 31 Oct 2025
Posts: 2
Given Kudos: 6
Posts: 2
Kudos: 0
If p and q are different prime numbers, m and n are integers, and pq 08 Dec 2024, 23:20
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi Bunuel, are there other problems of this type you can point me to? Got pretty stumped when seeing this problem type

Bunuel
guddo
If p and q are different prime numbers, m and n are integers, and pq is a divisor of mp + nq, which of the following must be true?

I. p is a divisor of n.
II. pq is a divisor of mp.
III. p^2 is a divisor of mn.

A. I only
B. II only
C. III only
D. I and II
E. I and III

Show Spoiler
Attachment:
2024-01-27_14-10-56.png

Given that pq is a divisor of mp + nq implies:


\(\frac{mp + nq}{pq} = integer\)

Splitting the fraction results in:


\(\frac{mp}{pq} + \frac{nq}{pq} = integer\)

\(\frac{m}{q} + \frac{n}{p} = integer\)

Since p and q are different prime numbers, we cannot have a case like 3/7 + 11/7 = 2. Therefore, for the sum to be an integer, both \(\frac{m}{q}\) and \(\frac{n}{p}\) must be integers. This means that q is a factor of m and p is a factor of n.

Let's evaluate the options:

I. p is a divisor of n.

As discussed above, this must be true.

II. pq is a divisor of mp.

The above says that \(\frac{mp}{pq}=\frac{m}{q}=integer\), and as discussed above, this must be true.

III. p^2 is a divisor of mn.

However, this is not necessarily true. For example, consider m = 2, n = 3, p = 3, and q = 2.

Answer: D.
Show more
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 19 Nov 2025
Posts: 105,390
Own Kudos:
778,336
Given Kudos: 99,977
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 105,390
Kudos: 778,336
If p and q are different prime numbers, m and n are integers, and pq 09 Dec 2024, 00:15
Kudos
Add Kudos
Bookmarks
Bookmark this Post
siddysir
Hi Bunuel, are there other problems of this type you can point me to? Got pretty stumped when seeing this problem type

Bunuel
guddo
If p and q are different prime numbers, m and n are integers, and pq is a divisor of mp + nq, which of the following must be true?

I. p is a divisor of n.
II. pq is a divisor of mp.
III. p^2 is a divisor of mn.

A. I only
B. II only
C. III only
D. I and II
E. I and III

Show Spoiler
Attachment:
2024-01-27_14-10-56.png

Given that pq is a divisor of mp + nq implies:


\(\frac{mp + nq}{pq} = integer\)

Splitting the fraction results in:


\(\frac{mp}{pq} + \frac{nq}{pq} = integer\)

\(\frac{m}{q} + \frac{n}{p} = integer\)

Since p and q are different prime numbers, we cannot have a case like 3/7 + 11/7 = 2. Therefore, for the sum to be an integer, both \(\frac{m}{q}\) and \(\frac{n}{p}\) must be integers. This means that q is a factor of m and p is a factor of n.

Let's evaluate the options:

I. p is a divisor of n.

As discussed above, this must be true.

II. pq is a divisor of mp.

The above says that \(\frac{mp}{pq}=\frac{m}{q}=integer\), and as discussed above, this must be true.

III. p^2 is a divisor of mn.

However, this is not necessarily true. For example, consider m = 2, n = 3, p = 3, and q = 2.

Answer: D.
Show more

Multiples and Factors problems: https://gmatclub.com/forum/search.php?s ... tag_id=185

Hope it helps.
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 19 Nov 2025
Posts: 16,267
Own Kudos:
76,999
Given Kudos: 482
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,267
Kudos: 76,999
If p and q are different prime numbers, m and n are integers, and pq 06 Jan 2025, 03:51
Kudos
Add Kudos
Bookmarks
Bookmark this Post
yrozenblum
If p and q are different prime numbers, m and n are integers, and pq is a divisor of mp + nq, which of the following must be true?

I. p is a divisor of n.
II. pq is a divisor of mp.
III. p^2 is a divisor of mn.

A. I only
B. II only
C. III only
D. I and II
E. I and III

Show Spoiler
Attachment:
2024-01-27_14-10-56.png
Show more

p and q are different prime numbers (say 2 and 3)

If pq is a divisor of mp + nq, it means p is a divisor of mp + nq and q is also a divisor of mp + nq.

If p is a divisor of mp + nq, we can see that mp is already divisible by p, so nq should also be divisible by p. Since q is a prime number, n MUST be divisible by p.
By the same logic, m MUST be divisible by q.

I. p is a divisor of n - As seen above, it must be true.

II. pq is a divisor of mp - Since m must be divisible by q, mp is divisible by pq. True.

III. p^2 is a divisor of mn - We know that p is a divisor of n for sure but we cannot say anything about p^2.

Answer (D)
User avatar
Juangui17
User avatar
Joined: 20 Apr 2024
Last visit: 20 Feb 2025
Posts: 9
Own Kudos:
4
Given Kudos: 72
Location: Colombia
Schools: Bocconi (A)
GPA: 3.57
Schools: Bocconi (A)
Posts: 9
Kudos: 4
If p and q are different prime numbers, m and n are integers, and pq 10 Jan 2025, 15:42
Kudos
Add Kudos
Bookmarks
Bookmark this Post
This question picking numbers is super simple, I hope I can build this confidence to do it in the exam because in the practice exam I had it wrong.

P=2
M=3
Q=3
N=2

(i) P/N = 2/2 TRUE
(ii) MP/PQ = 6/6 TRUE
(iii) MN/(P**2)=6/9 FALSE
Moderators:
Bunuel
Math Expert
105390 posts
Krunaal
Tuck School Moderator
805 posts