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• $450 Tuition Credit & Official CAT Packs FREE November 15, 2018 November 15, 2018 10:00 PM MST 11:00 PM MST EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) If x and y are positive integers and 1 + x + y +xy = 21  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: Hide Tags Manager Joined: 13 May 2011 Posts: 241 WE 1: IT 1 Yr WE 2: Supply Chain 5 Yrs If x and y are positive integers and 1 + x + y +xy = 21 [#permalink] Show Tags 11 Jun 2012, 07:20 2 8 00:00 Difficulty: 55% (hard) Question Stats: 62% (01:59) correct 38% (01:41) wrong based on 295 sessions HideShow timer Statistics If x and y are positive integers and 1 + x + y +xy = 21, what is the value of x? (1) y > 3 (2) y = 6 Most Helpful Expert Reply Math Expert Joined: 02 Sep 2009 Posts: 50544 Re: If x and y are positive integers and 1 + x + y +xy = 21 [#permalink] Show Tags 11 Jun 2012, 07:40 7 3 Good question. +1. If x and y are positive integers and 1 + x + y +xy = 21, what is the value of x? $$1 + x + y +xy = 21$$ --> $$(x+1)(y+1)=21$$ --> either $$x+1=3$$ and $$y+1=7$$ OR $$x+1=7$$ and $$y+1=3$$. Notice that $$x+1=1$$ and $$y+1=21$$ OR $$x+1=21$$ and $$y+1=1$$ is not possible since in this case either $$x$$ or $$y$$ equals zero and we are told that $$x$$ and $$y$$ are positive integers. Now, from $$x+1=3$$ and $$y+1=7$$ --> $$x=2$$ and $$y=6$$ AND from $$x+1=7$$ and $$y+1=3$$ --> $$x=6$$ and $$y=2$$. (1) y>3 --> $$y=6$$, so $$x=2$$. Sufficient. (2) y=6 --> $$x=2$$. Sufficient. Answer: D. Hope it's clear. _________________ General Discussion Current Student Joined: 28 Mar 2012 Posts: 316 Location: India GMAT 1: 640 Q50 V26 GMAT 2: 660 Q50 V28 GMAT 3: 730 Q50 V38 Re: If x and y are positive integers and 1 + x + y +xy = 21 [#permalink] Show Tags 11 Jun 2012, 07:29 2 Hi, We have, 1 + x + y +xy = 21 (where x & y are natural numbers/positive integers) or x(1+y) = 20 -y or x = (20-y)/(1+y) Using (1) y > 3, i.e., y = 4, 5, 6... when y=4, x = (20-4)/5 = 16/5 (not a natural number) when y=5, x = (20-5)/6 = 15/6 (not a natural number) when y=6, x = (20-6)/7 = 14/7 = 2 when y=7, x = (20-7)/8 = 13/8 (not a natural number), and for remaining values of y, there would be no positive integral value of x. so, (x,y) = (2,6). Sufficient Using (2) y = 6, => x = (20-6)/(1+6) = 2 so, (x,y) = (2,6). Sufficient. Thus, Answer is (D) Regards, Director Joined: 26 Oct 2016 Posts: 641 Location: United States Concentration: Marketing, International Business Schools: HBS '19 GMAT 1: 770 Q51 V44 GPA: 4 WE: Education (Education) Re: If x and y are positive integers and 1 + x + y +xy = 21 [#permalink] Show Tags 03 Mar 2017, 13:23 1 We can factor 1 + x + y + xy = 21 to determine that (1 + x)(1 + y) = 21. The product of the integers (1 + x) and (1 + y) is 21, which has the factors 1, 3, 7, and 21. The factor pair 1 and 21 is disqualified because neither (1 + x) nor (1 + y) could equal 1, as that would make one of the positive integers x or y equal to zero. We therefore determine that (1 + x) could equal 3 or 7, and conversely, (1 + y) could equal 7 or 3 such that their product is 21. Knowing that x will equal either 2 or 6, we can rephrase the question: “Is the value of x equal to 2 or 6?” (1) SUFFICIENT: If y > 3, then it must be true that y = 6 and x = 2. (2) SUFFICIENT: If y = 6, then x = 2. The correct answer is D. _________________ Thanks & Regards, Anaira Mitch Senior Manager Status: Come! Fall in Love with Learning! Joined: 05 Jan 2017 Posts: 489 Location: India Re: If x and y are positive integers and 1 + x + y +xy = 21 [#permalink] Show Tags 04 Mar 2017, 05:06 Prompt analysis x an y are the integers such that 1 +x +y +xy =21 or (1+x)(1+y) = 21 therefore (1+x)(1+y) could be 1 x 21. hence x =0, y = 20 (1) 3 x 7. hence x =2, y = 6 (2) 7 x 3. hence x =6, y = 2 (3) 21 x 1. . hence x =20, y = 0 (4) Translation To find the exact solution among the 4 options we need 1# the range of x and y 2# exact value of x and y Statement analysis St 1: x>3. hence option (3) and (4) applicable. INSUFFICIENT St 2: y =6. Hence option (2) only is applicable. ANSWER Option B _________________ GMAT Mentors Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6497 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If x and y are positive integers and 1 + x + y +xy = 21 [#permalink] Show Tags 18 Mar 2018, 21:59 BDSunDevil wrote: If x and y are positive integers and 1 + x + y +xy = 21, what is the value of x? (1) y>3 (2) y=6 [Searched, but couldn't find] Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. 1 + x + y +xy = 21 ⇔ (x+1)(y+1) = 21 ⇔ x+1 = 3, y+1 = 7 or x+1 = 7, y+1 = 3 since x and y are positive integers. ⇔ x=2, y=6 or x=6, y=2 Condition 1) y > 3 ⇒ y = 6 ⇒ x = 2 The condition 1) is sufficient. Condition 2) y = 6 ⇒ x = 2 The condition 2) is sufficient. Therefore, D is the answer. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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If x and y are positive integers and 1 + x + y +xy = 21  [#permalink]

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Updated on: 09 Sep 2018, 00:34
Factor 1 + x + y + xy = 21 to determine that (1 + x)(1 + y) = 21. The product of the integers (1 + x) and (1 + y) is 21, which has the factors 1, 3, 7, and 21. The factor pair 1 and 21 is disqualified because neither (1 + x) nor (1 + y) could equal 1, as that would make x or y equal to zero (and they must be positive integers). Therefore, (1 + x) could equal 3 or 7, and conversely, (1 + y) could equal 7 or 3 such that their product is 21.

Knowing that x will equal either 2 or 6, rephrase the question: “Is the value of x equal to 2 or 6?”

(1) SUFFICIENT: If y > 3, then it must be true that y = 6 and x = 2.

(2) SUFFICIENT: If y = 6, then x = 2.

Originally posted by alphak3nny001 on 08 Sep 2018, 20:35.
Last edited by Bunuel on 09 Sep 2018, 00:34, edited 2 times in total.
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If x and y are positive  [#permalink]

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08 Sep 2018, 21:01
alphak3nny001 wrote:
If x and y are positive integers and 1 + x + y + xy = 21, what is the value of x?

(1) y > 3
(2) y = 6

Given, $$(x+1)(y+1) = 21 = 3*7$$

Statement I:

For $$(y+1)(x+1) = 21$$ to satisfy $$y = 6, x = 2$$ (As x, y are positive Integer).

Statement II:

$$y = 6, x = 2$$

Hence, D.
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If x and y are positive  [#permalink]

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08 Sep 2018, 21:54
alphak3nny001 wrote:
If x and y are positive integers and 1 + x + y + xy = 21, what is the value of x?

(1) y > 3
(2) y = 6

Two ways...

1) Algebraic..
$$1+x+y+xy=(1+x)+y(1+x)=(1+y)(1+x)=21$$
Now since X and y are positive integers so X+1 and y+1 will be greater than 1..
Thus 21=1*21 not possible
Only other possibility is 21=3*7..
Thus (1+x)(1+y)=3*7...
So x and y are 3-1=2 and 7-1=6

We have to find which value is which variable..

1) y>3
Therefore y cannot be 2, X is 2
Sufficient

2) y=6
Thus X is 2
Sufficient

D

2) Trial

1+x+y+xy=21.....x+y+xy=20
1) y>3
So least value of y is 4, substitute
x+4+4x=20.....5x=16 but then x is not an integer
Let y=5.....5+x+5x=20....6x=15...NO
Let y =6.....6+x+6x=20....7x=14...X=2... possible
As we increase y X decrease, so lowest possible of X is 1..
So 1+y+y=20....2y=19...not possible
Thus X=2
Sufficient
2) y=6
1+6+x+6x=21.......X=2
Sufficient

D
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3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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If x and y are positive &nbs [#permalink] 08 Sep 2018, 21:54
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