Forum Home > GMAT > Quantitative > Problem Solving (PS)
Events & Promotions
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
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Jul 2710:00 AM PDT
-11:00 AM PDT
Are you struggling with the GMAT verbal section? Do tricky questions and lengthy passages overwhelm you? Don't worry—I've got you covered! In this video, I'll provide a detailed, actionable plan to help you significantly boost your verbal score.
Jul 2508:00 AM EDT
-11:59 PM EDT
More and more TTP students are earning impressive scores on the GMAT. Now is the perfect time to join the many GMAT students who made the switch to TTP and surpassed their wildest expectations on test day.- Jul 28
10:00 AM PDT
-11:00 AM PDT
We will learn how to guess an answer choice better on the GMAT. If you are running out of time and want to guess on a few questions, these effective techniques will help you in making a smart guess. 
Jul 2811:00 AM IST
-01:00 PM IST
Attend this free GMAT Algebra Webinar and learn how to master the most challenging Inequalities and Absolute Value problems with ease.- Jul 31
05:55 AM PDT
-12:30 PM PDT
MBA Application Walkthrough and Essay Review. You will have a chance to hear from experts covering the Top 15 MBA programs, covering details of the application and essays. 
Jul 3110:00 AM EDT
-11:59 PM PDT
Save 20% on Target Test Prep and Score in the 100th Percentile in GMAT Verbal! Our course uses techniques such as topical study and spaced repetition to maximize knowledge retention and make studying simple and fun.
If p and q are both positive integers, then is p divisible by 9?
[#permalink]
05 Mar 2019, 16:02
05 Mar 2019, 16:02
1
Kudos
5
Bookmarks
Show timer
00:00
A
B
C
D
E
Difficulty:
45%
(medium)
Question Stats:
65% (01:31) correct
35%
(01:54)
wrong
based on 128
sessions
If p and q are both positive integers, then is p divisible by 9?
1) p/10 + q/10 is an integer.
2) p/9 + q/10 is an integer.
1) p/10 + q/10 is an integer.
2) p/9 + q/10 is an integer.
Re: If p and q are both positive integers, then is p divisible by 9?
[#permalink]
05 Mar 2019, 20:02
05 Mar 2019, 20:02
5
Kudos
3
Bookmarks
Expert Reply
If p and q are both positive integers, then is p divisible by 9?
1) \(\frac{p}{10} + \frac{q}{10}\) is an integer.
Various ways possible..
If p=q=5, \(\frac{p}{10} + \frac{q}{10}=1\) , but p or 5 is not divisible by 9.
If p=9 and q=1, \(\frac{p}{10} + \frac{q}{10}=1\) , but p or 9 is divisible by 9.
Thus, insufficient
2) \(\frac{p}{9} + \frac{q}{10}\) is an integer.
Now the denominator are different in both cases, so let us solve
\(\frac{p}{9} + \frac{q}{10}=x....10p+9q=90x...10p=90x-9q=9(10x-q)\)
Now 10p is surely a multiple of 9, so only way i sthat p is a multiple of 9
Sufficient
B
1) \(\frac{p}{10} + \frac{q}{10}\) is an integer.
Various ways possible..
If p=q=5, \(\frac{p}{10} + \frac{q}{10}=1\) , but p or 5 is not divisible by 9.
If p=9 and q=1, \(\frac{p}{10} + \frac{q}{10}=1\) , but p or 9 is divisible by 9.
Thus, insufficient
2) \(\frac{p}{9} + \frac{q}{10}\) is an integer.
Now the denominator are different in both cases, so let us solve
\(\frac{p}{9} + \frac{q}{10}=x....10p+9q=90x...10p=90x-9q=9(10x-q)\)
Now 10p is surely a multiple of 9, so only way i sthat p is a multiple of 9
Sufficient
B
General Discussion
Re: If p and q are both positive integers, then is p divisible by 9?
[#permalink]
06 Mar 2019, 09:01
06 Mar 2019, 09:01
selim wrote:
If p and q are both positive integers, then is p divisible by 9?
1) p/10 + q/10 is an integer.
2) p/9 + q/10 is an integer.
1) p/10 + q/10 is an integer.
2) p/9 + q/10 is an integer.
IF YOU FIND MY SOLUTION TO BE HELPFUL, PLEASE GIVE ME KUDOS
(1) p/10 + q/10 is an integer
Let p =8, q = 2, This satisfies our constraint and p is not divisible by 9.
Let p =9 and q = 1. This satisfies our constraint and p is divisible by 9 NS
(2) p/9 +q/10 is an integer.
So, our target denominator for the sum of our terms is going to be 90 (we will have to convert p/9 + q/10 to a common denominator of 90). This results in q being multiplied by 9 and p being multiplied by 10 to convert to the LCD of 90.
Then our q term will contribute one of the following values to the numerator of our sum 9(1), 9(2), 9(3)...= 9,18,27,36,45,54,63,72,81,90....
Our p term will be multiplied by 10 when converting to our common denominator. So, our p term will contribute one of the following values to our final sum:
10,20,30,40,50,60,70,80,90,.... Notice, that our sum will only be divisible by 90, if both 10p and 9q are both multiples of 90, therefore p is a multiple of 9 sufficient.
