Last visit was: 22 Apr 2026, 11:03 It is currently 22 Apr 2026, 11:03
Home
Messages
Tests
Schools
Events
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
805+ (Hard)|   Algebra|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 22 Apr 2026
Posts: 109,748
Own Kudos:
810,643
 [11]
Given Kudos: 105,821
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: 109,748
Kudos: 810,643
 [11]
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 09 Dec 2021, 08:45
2
Kudos
Add Kudos
8
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:

95% (hard)

Question Stats:

53% (02:40) correct 47% (02:47) wrong based on 211 sessions

History

Date
Time
Result
Not Attempted Yet
How many pairs of positive integers (x, y) satisfy the equation 3x + 2y = 120 such that y > x.

A. 7
B. 8
C. 10
D. 11
E. 12



Are You Up For the Challenge: 700 Level Questions
ShowHide Answer
Official Answer
D
avatar
Anurocks1233
avatar
Joined: 01 Oct 2021
Last visit: 25 Oct 2022
Posts: 44
Own Kudos:
64
 [1]
Given Kudos: 90
Posts: 44
Kudos: 64
 [1]
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 14 Dec 2021, 11:09
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
X= 40 - 2/3 y
Y =3 to 60 so total 20 integers are possible.
10 cases where y>x
User avatar
StacyArko
User avatar
Joined: 24 Jan 2020
Last visit: 22 Apr 2026
Posts: 23
Own Kudos:
7
 [2]
Given Kudos: 156
Location: Ghana
GMAT 1: 600 Q39 V34
Products:
Forum Quiz
GMAT 1: 600 Q39 V34
Posts: 23
Kudos: 7
 [2]
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 14 Dec 2021, 17:06
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Using the exchange rate method, x=y= 24 satisfies.

Exchange x (@-2) and y (@+3) so y>x satisfies. When x is reduced by 2, we get 22, 20, 18.... 2 (giving 11 pairs).

IMO: D

Posted from my mobile device
User avatar
SaquibHGMATWhiz
User avatar
GMATWhiz Representative
Joined: 23 May 2022
Last visit: 12 Jun 2024
Posts: 623
Own Kudos:
777
 [1]
Given Kudos: 6
Location: India
GMAT 1: 760 Q51 V40
Expert
Expert reply
GMAT 1: 760 Q51 V40
Posts: 623
Kudos: 777
 [1]
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 05 Feb 2023, 05:32
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Bunuel
How many pairs of positive integers (x, y) satisfy the equation 3x + 2y = 120 such that y > x.

A. 7
B. 8
C. 10
D. 11
E. 12
Show more

Solution:

  • Given equation \(3x+2y=120\)
    \(⇒2y=120-3x\)
    \(⇒y=\frac{120-3x}{2}\)
    \(⇒y=60-\frac{3x}{2}\)
  • Since x and y are positive integers, x has to be even number
  • If \(x=2\) (minimum value), then \(y=60-\frac{3\times 2}{2}=60-3=57\)

  • So, \(x=2\) and \(y=57\) is one of the value
  • For other possible values, we can add values of x with the coefficient of y i.e., 2 and value of y with the coefficient of x i.e., 3
    • This is the concept of special equation which we have discussed at length in this blog
    Attachment:
    special1.png
    special1.png [ 8.52 KiB | Viewed 3797 times ]
  • Since y has to be greater than x, there are total 11 possible pairs of possible values

Hence the right answer is Option D
User avatar
Ashwi
User avatar
Joined: 21 Sep 2016
Last visit: 22 Jul 2025
Posts: 5
Own Kudos:
2
 [2]
Given Kudos: 441
Posts: 5
Kudos: 2
 [2]
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 08 Jun 2025, 23:23
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
We are given the linear equation: 3x+2y=120,

Step 1: Solve for x
x= ( 120−2y ) / 3 => x = 60 - 2/3 y

Since we are only interested in positive integer pairs, both x>0 and y>0

To keep x >0,
( 120−2y ) / 3 > 0 => 0< y < 60

Step 2: Apply the constraint y>x
y > (120−2y) / 3 => y > 24

Now combine both conditions for y: 60> y > 24

Now, from step 1 expression
For x to be an integer, y must be multiple of 3

So valid y values between 24 and 60 that are divisible by 3 are:
y=27,30,...,57

This is an arithmetic sequence. Count the terms = ( 57 - 27 ) / 3 + 1 = 11

There are 11 values of y that satisfy all conditions, so there are 11 valid (x, y) pairs.
User avatar
eb590
User avatar
Joined: 23 Jan 2025
Last visit: 16 Feb 2026
Posts: 13
Own Kudos:
3
Given Kudos: 4
Posts: 13
Kudos: 3
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 13 Jun 2025, 03:00
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Isn't x = 40 - 2/3y ?

Ashwi
We are given the linear equation: 3x+2y=120,

Step 1: Solve for x
x= ( 120−2y ) / 3 => x = 60 - 2/3 y

Since we are only interested in positive integer pairs, both x>0 and y>0

To keep x >0,
( 120−2y ) / 3 > 0 => 0< y < 60

Step 2: Apply the constraint y>x
y > (120−2y) / 3 => y > 24

Now combine both conditions for y: 60> y > 24

Now, from step 1 expression
For x to be an integer, y must be multiple of 3

So valid y values between 24 and 60 that are divisible by 3 are:
y=27,30,...,57

This is an arithmetic sequence. Count the terms = ( 57 - 27 ) / 3 + 1 = 11

There are 11 values of y that satisfy all conditions, so there are 11 valid (x, y) pairs.
Show more
User avatar
Vittal123
User avatar
Joined: 06 May 2024
Last visit: 15 Mar 2026
Posts: 3
Given Kudos: 51
Posts: 3
Kudos: 0
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 02 Nov 2025, 17:18
Kudos
Add Kudos
Bookmarks
Bookmark this Post
StacyArko
Using the exchange rate method, x=y= 24 satisfies.

Exchange x (@-2) and y (@+3) so y>x satisfies. When x is reduced by 2, we get 22, 20, 18.... 2 (giving 11 pairs).

IMO: D

Posted from my mobile device
Show more
Hey stacy, this looks like a nifty trick. Could you elaborate on this exchnage rate method.
User avatar
Wadree
User avatar
Joined: 06 Sep 2024
Last visit: 07 Jan 2026
Posts: 68
Own Kudos:
2
Given Kudos: 230
Posts: 68
Kudos: 2
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 06 Nov 2025, 08:32
Kudos
Add Kudos
Bookmarks
Bookmark this Post
3X + 2Y = 120 ........ Y>X ....... X and Y integer.......
Now, Y has 2 multiplied with it.... So 120 - 3X must be even so that when we divide 120 - 3X by 2 (another even).....the result is even and integer...... So 120 - 3X must be even......now, 120 is even..... So 3X must be even so that when we subtract 3X from 120.....the result is even ....... So, 3X must be even...... Then X must be even because 3 is odd....and (odd)×(odd) = (odd)......(odd)×(even) = (even)........so X must be even.......so least positive even integer is 2..... So if we plug X = 2......then Y = 57.....now we gotta test X = 2,4,6......but the options in the question say highest 12 values for X and Y......so if X = 2,4,6...... Then 12th value for X must be 24.....but if X = 24...... Then Y = 24.....but Y ≠ X so X cant have 12th value....so X has 11 possible values.....then Y also has 11 possible corresponding values......so answer is 11......
User avatar
Kinshook
User avatar
Major Poster
Joined: 03 Jun 2019
Last visit: 22 Apr 2026
Posts: 5,986
Own Kudos:
5,858
Given Kudos: 163
Location: India
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Products:
My Reviews
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
Posts: 5,986
Kudos: 5,858
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 06 Nov 2025, 18:13
Kudos
Add Kudos
Bookmarks
Bookmark this Post
How many pairs of positive integers (x, y) satisfy the equation 3x + 2y = 120 such that y > x.

y = (120-3x)/2 = 60 - 1.5x

x = 2; y = 57
x = 4; y = 54
x = 6; y = 51
x = 8; y = 48
x = 10; y = 45
x = 12; y = 42
x = 14; y = 39
x = 16; y = 36
x = 18; y = 33
x = 20; y = 30
x = 22; y = 27
x = 24; y = 24 ; Not valid since y > x

11 pairs

IMO D
User avatar
Kinshook
User avatar
Major Poster
Joined: 03 Jun 2019
Last visit: 22 Apr 2026
Posts: 5,986
Own Kudos:
5,858
Given Kudos: 163
Location: India
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Products:
My Reviews
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
Posts: 5,986
Kudos: 5,858
How many pairs of positive integers (x, y) satisfy the equation 3x + 2 06 Nov 2025, 18:15
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Another way to solve

x = 2k
y = 60 - 3k

y > x ; 60 - 3k > 2k; 5k < 60 ; k < 12

k = 1 to 11 ; 11 pairs

Kinshook
How many pairs of positive integers (x, y) satisfy the equation 3x + 2y = 120 such that y > x.

y = (120-3x)/2 = 60 - 1.5x

x = 2; y = 57
x = 4; y = 54
x = 6; y = 51
x = 8; y = 48
x = 10; y = 45
x = 12; y = 42
x = 14; y = 39
x = 16; y = 36
x = 18; y = 33
x = 20; y = 30
x = 22; y = 27
x = 24; y = 24 ; Not valid since y > x

11 pairs

IMO D
Show more
Moderators:
Bunuel
Math Expert
109748 posts
Krunaal
Tuck School Moderator
853 posts