GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

It is currently 29 Jan 2020, 04:58

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.

Close

Request Expert Reply

Confirm Cancel

Let p and q be two digit integers such that q is obtained by reversing

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 60778
Let p and q be two digit integers such that q is obtained by reversing  [#permalink]

Show Tags

New post 04 Dec 2019, 02:12
1
8
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

50% (03:02) correct 50% (03:33) wrong based on 20 sessions

HideShow timer Statistics

Let p and q be two digit integers such that q is obtained by reversing the digits of p. The integers p and q satisfy the equation \(p^2 - q^2 = r^2\) for some positive integer r. what is the value of \(p+q+r\) ?

(A) 88
(B) 112
(C) 116
(D) 144
(E) 154

Are You Up For the Challenge: 700 Level Questions

_________________
GMAT Club Legend
GMAT Club Legend
User avatar
V
Joined: 18 Aug 2017
Posts: 5751
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
GMAT ToolKit User Premium Member
Re: Let p and q be two digit integers such that q is obtained by reversing  [#permalink]

Show Tags

New post 04 Dec 2019, 02:56
1
p=10a+b
q=10b+a
\(p^2 - q^2 = r^2\)
we get
(11a+11b)(9a-9b)=r^2
r= 3*√11*(a+b)*(a-b)
a+b has to be 11 ; 6+5 ;
so r = 33
no ; 65+56+33 ; 154
IMO E

Bunuel wrote:
Let p and q be two digit integers such that q is obtained by reversing the digits of p. The integers p and q satisfy the equation \(p^2 - q^2 = r^2\) for some positive integer r. what is the value of \(p+q+r\) ?

(A) 88
(B) 112
(C) 116
(D) 144
(E) 154

Are You Up For the Challenge: 700 Level Questions
Manager
Manager
User avatar
S
Joined: 03 Nov 2019
Posts: 54
CAT Tests
Re: Let p and q be two digit integers such that q is obtained by reversing  [#permalink]

Show Tags

New post 04 Dec 2019, 05:43
p= 10A+B ; q=10B+A

Also A and B are first digit of 2 digit numbers so they cannot be zero hence A and B lies between 1 and 9.

p+q=11*(A+B)

p-q=9*(A-B) ........................................consider all the four statement above as eq (i)



r^2= p^2-q^2 = (p+q)(p-q) =99(A+B)(A-B)

r is a positive integer therefore r cannot be 0 or A is not equal to B and A>B

and maximum possible value of A-B would be 9-1 =8................................consider above two statement above as eq (iii)



Now for 99(A+B)(A-B) should be a square term

so minimum possible value of (A+B)(A-B)=11

And since 11 is prime so either (A+B) =11 or (A-B)=11

But from iii A+B = 11 therefore A-B= 1



thus r is a multiple of 11*3=33

r = 33, 66, 99, 132, 165, 198 ....=33K ...K=some constant



Minimum value of

p+q+r= 11(A+B)+33 .............(ii)

=11*11+33= 121+33 = 154

Answer : E

Posted from my mobile device
VP
VP
avatar
P
Joined: 24 Nov 2016
Posts: 1143
Location: United States
CAT Tests
Re: Let p and q be two digit integers such that q is obtained by reversing  [#permalink]

Show Tags

New post 06 Dec 2019, 07:59
Bunuel wrote:
Let p and q be two digit integers such that q is obtained by reversing the digits of p. The integers p and q satisfy the equation \(p^2 - q^2 = r^2\) for some positive integer r. what is the value of \(p+q+r\) ?

(A) 88
(B) 112
(C) 116
(D) 144
(E) 154


p=10a+b, q=10b+a
r^2=p^2-q^2, r^2=(p+q)(p-q)
r^2=(10a+b+(10b+a))(10a+b-(10b+a))
r^2=(11a+11b)(9a-9b), r^2=11*9*(a+b)(a-b)
r^2 = positive integer, (a+b)(a-b)>0, 9≥a>b>0

9 = perf square
11*(a+b)(a-b) = perf square
(a+b) = multiple of 11 because 8≥(a-b)≥0
(a,b) = [9,2; 8,3; 7,4; 6,5]
(a-b) = perf square
(a,b) = [6,5]

r^2=11*9*11*1=33
p=65, q=56, r=33, p+q+r=154

Ans (E)
GMAT Club Bot
Re: Let p and q be two digit integers such that q is obtained by reversing   [#permalink] 06 Dec 2019, 07:59
Display posts from previous: Sort by

Let p and q be two digit integers such that q is obtained by reversing

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne