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Math Expert V
Joined: 02 Sep 2009
Posts: 64096
The positive two-digit integers x and y have the same digits, but in  [#permalink]

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6
29 00:00

Difficulty:   5% (low)

Question Stats: 87% (00:54) correct 13% (01:12) wrong based on 1651 sessions

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The positive two-digit integers x and y have the same digits, but in reverse order. Which of the following must be a factor of x + y?

(A) 6
(B) 9
(C) 10
(D) 11
(E) 14

Kudos for a correct solution.

_________________
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 16711
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: The positive two-digit integers x and y have the same digits, but in  [#permalink]

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7
5
Hi All,

This question can be solved by TESTing VALUES:

We're told that X and Y are both 2-digit positive integers with the digits reversed. We're asked for what MUST be a factor of X+Y

IF....
X=12
Y=21

X+Y = 12+21 = 33

Only one of the answers is a factor of 33...

GMAT assassins aren't born, they're made,
Rich
_________________
Current Student B
Joined: 10 Mar 2013
Posts: 449
Location: Germany
Concentration: Finance, Entrepreneurship
Schools: WHU MBA"20 (A$) GMAT 1: 580 Q46 V24 GPA: 3.88 WE: Information Technology (Consulting) Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags 11 6 Bunuel wrote: The positive two-digit integers x and y have the same digits, but in reverse order. Which of the following must be a factor of x + y? (A) 6 (B) 9 (C) 10 (D) 11 (E) 14 Kudos for a correct solution. 10a+b+10b+a=11(a+b) Answer (D) ##### General Discussion SVP  V Status: It's near - I can see. Joined: 13 Apr 2013 Posts: 1685 Location: India Concentration: International Business, Operations Schools: INSEAD Jan '19 GPA: 3.01 WE: Engineering (Real Estate) Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags 4 4 Bunuel wrote: The positive two-digit integers x and y have the same digits, but in reverse order. Which of the following must be a factor of x + y? (A) 6 (B) 9 (C) 10 (D) 11 (E) 14 Kudos for a correct solution. My Solution: Lets say integer x has two digits "ab" then as per question y has two digits in reverse i.e, "ba" , Then x+y will be, (10a+b)+(10b+a)----> 11a+11b---->11(a+b) Therefore 11 must be a factor of x+y Answer D _________________ "Do not watch clock; Do what it does. KEEP GOING." Manager  Status: tough ... ? Naaahhh !!!! Joined: 07 Sep 2015 Posts: 61 Location: India Concentration: Marketing, Strategy WE: Marketing (Computer Hardware) Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags 2 10a+b+10b+a 11a+11b 11(a+b)....Ans: 11 Intern  S Joined: 12 Nov 2014 Posts: 18 Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags WHy not 3,6. 36 and 63 are both divisible by 9. EMPOWERgmat Instructor V Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 16711 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags 1 Hi alice7, The prompt asks us for what MUST be a factor of X+Y... Using your example (36 and 63), we would have a total of 99. In this case, TWO of the answers 'fit' - both 9 and 11 are factors of 99. So one of these MUST be the solution, but we won't know which one until we find another example that is NOT divisible by one of the two options. GMAT assassins aren't born, they're made, Rich _________________ Manager  Joined: 06 Jun 2014 Posts: 84 Location: United States Concentration: Finance, General Management GMAT 1: 450 Q27 V21 GPA: 3.47 Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags 1 2 Bunuel wrote: The positive two-digit integers x and y have the same digits, but in reverse order. Which of the following must be a factor of x + y? (A) 6 (B) 9 (C) 10 (D) 11 (E) 14 Kudos for a correct solution. Remember: When you take the difference between the two, it will always be 9. e.g 23-32=9, 89-98=9 and when you add both integers, the sum will always be a multiple of 11 e.g 23+32=55, 89+98= 187 so the answer is 11 Manager  Joined: 09 Jun 2015 Posts: 78 Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags 1 Bunuel wrote: The positive two-digit integers x and y have the same digits, but in reverse order. Which of the following must be a factor of x + y? (A) 6 (B) 9 (C) 10 (D) 11 (E) 14 Kudos for a correct solution. Supposing that the numbers are ab and ba, then ab can be written as 10a+b and ba can be written as 10b+a Adding we get 11a+11b=11*(a+b) Therefore, the sum is divisible by 11 as well as by sum of the digits. Math Expert V Joined: 02 Aug 2009 Posts: 8586 Re: The positive two-digit integers x and y [#permalink] ### Show Tags 1 macbookno wrote: The positive two-digit integers x and y have the same digits, but in reverse order. Which of the following must be a factor of x + y? A. 6 B. 9 C. 10 D. 11 E. 14 Pl post according to guidelines.. Provide OA, proper topic name and post in correct sub-forum.. as for your Q.. let x = ab, a 2-digit integer so y= ba.. x= ab = 10a+b and y=ba=10b+a.. so $$x+y = 10a+b+10b+a = 11(a+b)$$.. therefore x+y will always have a factor 11.. D _________________ Veritas Prep GMAT Instructor V Joined: 16 Oct 2010 Posts: 10441 Location: Pune, India Re: The positive two-digit integers x and y [#permalink] ### Show Tags 1 macbookno wrote: The positive two-digit integers x and y have the same digits, but in reverse order. Which of the following must be a factor of x + y? A. 6 B. 9 C. 10 D. 11 E. 14 Take a simple example. Two integers could be 12 and 21. x + y = 33. Only 11 is a factor. So 11 must be a factor of (x+y) Answer (D) _________________ Karishma Veritas Prep GMAT Instructor Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options > Target Test Prep Representative V Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 10543 Location: United States (CA) Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags 1 2 Bunuel wrote: The positive two-digit integers x and y have the same digits, but in reverse order. Which of the following must be a factor of x + y? (A) 6 (B) 9 (C) 10 (D) 11 (E) 14 Kudos for a correct solution. We can solve this question using the natural relationships that all two-digit numbers have. As an example, we can express 37 as (10 x 3) + 7. We multiply the digit in the tens position by 10 and then add the digit in the ones position. If we let a = the tens digit of x and b = the ones digit of x, we know: x = 10a + b Since the digits of y are the reverse of those of x, we can express y as: y = 10b + a When we sum x and y we obtain: x + y = 10a + b + 10b + a = 11a + 11b x + y = 11(a + b) The final expression 11(a + b) is a multiple of 11, and therefore 11 divides evenly into it. We see, therefore, that 11 must be a factor of x + y. Answer: D _________________ # Scott Woodbury-Stewart Founder and CEO Scott@TargetTestPrep.com See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Manager  Joined: 20 Mar 2015 Posts: 52 The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags EMPOWERgmatRichC wrote: Hi All, This question can be solved by TESTing VALUES: We're told that X and Y are both 2-digit positive integers with the digits reversed. We're asked for what MUST be a factor of X+Y IF.... X=12 Y=21 X+Y = 12+21 = 33 Only one of the answers is a factor of 33... Final Answer: GMAT assassins aren't born, they're made, Rich I thought same digits were 11, 22, 33, 44, 55, 66..... 99. I am confused now. Board of Directors D Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4989 Location: India GPA: 3.5 WE: Business Development (Commercial Banking) Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags bimalr9 wrote: I thought same digits were 11, 22, 33, 44, 55, 66..... 99. I am confused now. Ok Let me try once - X = 14 , Y = 41 X + Y = 14 + 41 =>55 55 = 11 * 5 Check again - X = 12 , Y = 21 X + Y = 12 + 21 =>33 33 = 11 * 3 Check again - X = 16 , Y = 61 X + Y = 16 + 61 =>77 77 = 11 * 7 Check in each case the common factor is 11 , hence the answer must be 11...... Feel free to revert in case of the slightest doubt, I will love to explain it again... PS: IMHO the best method for this problem will be (10a + b ) + (10b + a ) => 11(a+b) as posted earlier... _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only ) Current Student D Joined: 12 Aug 2015 Posts: 2522 Schools: Boston U '20 (M) GRE 1: Q169 V154 Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags Here is my solution for this one => x=MN => 10M+N y=NM=>10N+M x+y=> Must be a multiple of 11 OR in other words => 11 must be a factor of x+y Alternatively Let x=23 y=32 x+y=55 Only option that is a factor of 55 is 11 i.e Option D Hence D Additionally x-y will always be multiple of 9  _________________ IIMA, IIMC School Moderator V Joined: 04 Sep 2016 Posts: 1423 Location: India WE: Engineering (Other) The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags VeritasPrepKarishma Bunuel Engr2012 Quote: Take a simple example. Two integers could be 12 and 21. Is not the question stem ambiguous, leaving scope for discrepancy ? Ideal Q should be : no of digits are the same if no if digits of x and y are same, x = y = 10x + y Is q says reversing digits makes x and y same, it should be inferred as 10x + y = 10 y + x Let me know your understanding. Intern  B Joined: 20 Sep 2016 Posts: 19 Re: The positive two-digit integers x and y have the same digits, but in [#permalink] ### Show Tags take x as 23 and y as 32 x+y= 55 prime factors of 55 is 5 and 11. thus 11 is the answer! Intern  B Joined: 01 Mar 2019 Posts: 34 Location: United States Schools: Owen '22 (M$)
Re: The positive two-digit integers x and y have the same digits, but in  [#permalink]

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 Set Up Equation for X and Y

X = 10a + b

Y = 10b + a

X + Y = 10a + a + 10b + b

X + Y = 11a + 11b

X + Y = 11(a + b)

Intern  Joined: 13 Apr 2019
Posts: 1
Re: The positive two-digit integers x and y have the same digits, but in  [#permalink]

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roundh0use wrote:
 Set Up Equation for X and Y

X = 10a + b

Y = 10b + a

X + Y = 10a + a + 10b + b

X + Y = 11a + 11b

X + Y = 11(a + b)

why do we take 10 instead of any other number? for eg if we use 20, we end up with 21(a+b) and 21 is not one of the answer choices.
Non-Human User Joined: 09 Sep 2013
Posts: 14969
Re: The positive two-digit integers x and y have the same digits, but in  [#permalink]

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_________________ Re: The positive two-digit integers x and y have the same digits, but in   [#permalink] 19 May 2020, 07:56

# The positive two-digit integers x and y have the same digits, but in  