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# xy and yx are a pair of two digit positive integers with reversed digi

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VP
Joined: 07 Dec 2014
Posts: 1153
xy and yx are a pair of two digit positive integers with reversed digi  [#permalink]

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11 Nov 2017, 16:11
4
00:00

Difficulty:

65% (hard)

Question Stats:

51% (02:11) correct 49% (02:40) wrong based on 69 sessions

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xy and yx are a pair of two digit positive integers with reversed digits. For how many such pairs does yx-xy=y^2-x^2?

A. 2
B. 3
C. 4
D. 5
E. 6
Math Expert
Joined: 02 Aug 2009
Posts: 7334
xy and yx are a pair of two digit positive integers with reversed digi  [#permalink]

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11 Nov 2017, 18:53
2
gracie wrote:
xy and yx are a pair of two digit positive integers with reversed digits. For how many such pairs does yx-xy=y^2-x^2?

A. 2
B. 3
C. 4
D. 5
E. 6

Hi..

=$$yx-xy=y^2-x^2...........10y+x-10x-y+(x-y)(x+y)=0.......9(x-y)+(x-y)(x+y)=0.. (y-x)(9-(x+y)=0$$
Either x=y or x+y=9.... But pair means x and y should be different otherwise pair would consist of just one number example 22
So numbers with sum of digits as 9 are
18 and 81
27 and 72
36 and 63
45 and 54

Since pair of numbers is asked 18 and 81 will be same as 81 and 18.

C
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

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xy and yx are a pair of two digit positive integers with reversed digi  [#permalink]

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11 Nov 2017, 19:19
2
1
gracie wrote:
xy and yx are a pair of two digit positive integers with reversed digits. For how many such pairs does yx-xy=y^2-x^2?

A. 2
B. 3
C. 4
D. 5
E. 6

$$yx-xy=y^2-x^2$$

$$=>10y+x-10x-y=(y-x)(y+x) => 9y-9x-(y-x)(y+x)=0$$

or $$(y-x)[9-(y+x)]=0 =>$$ either $$y=x$$ or $$y+x=9$$

Now it is given that $$xy$$ & $$yx$$ are a pair of two digit integers, hence $$y$$ is not equal to $$x$$ because in that case there will be only a single no and not a pair

so $$y+x=9$$ the pairs possible are

(1,8), (2,7), (3,6), (4,5) = $$4$$.........[here (4,5) & (5,4) represent the same pair]

Option C
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Re: xy and yx are a pair of two digit positive integers with reversed digi  [#permalink]

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05 Jan 2019, 20:14
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: xy and yx are a pair of two digit positive integers with reversed digi   [#permalink] 05 Jan 2019, 20:14
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