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

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

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New post 11 Nov 2017, 17:11
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  65% (hard)

Question Stats:

47% (01:30) correct 53% (01:26) wrong based on 36 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
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Posts: 6549
xy and yx are a pair of two digit positive integers with reversed digi  [#permalink]

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New post 11 Nov 2017, 19:53
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



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


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

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New post 11 Nov 2017, 20:19
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
xy and yx are a pair of two digit positive integers with reversed digi &nbs [#permalink] 11 Nov 2017, 20:19
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