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# xy and yx are reversed two digit positive integers. If the difference

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VP
Joined: 07 Dec 2014
Posts: 1223
xy and yx are reversed two digit positive integers. If the difference  [#permalink]

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29 Sep 2017, 17:37
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6
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Difficulty:

35% (medium)

Question Stats:

77% (02:04) correct 23% (01:49) wrong based on 107 sessions

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xy and yx are reversed two digit positive integers. If the difference between them is a perfect square>10, what is the difference between x and y?

A. 2
B. 3
C. 4
D. 5
E. 6
Manager
Joined: 12 Dec 2016
Posts: 77
Re: xy and yx are reversed two digit positive integers. If the difference  [#permalink]

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29 Sep 2017, 19:01
5
C. x and y are place holder values. Let the first number be A and the second number be B.

A = 10x+y, B= 10y+x

(for example 57 = 10(5) +7(1), 957 = 100(9)+10(5)+7(1))

A-B = 10x+y-(10y+x)

= 10x+y-10y-x

= 10x-x+y-10y

=9x-9y

We are told 9x-9y >10 and is a perfect square

9x-9y = 9(x-y)

= 3^2( x-y)

x-y must also be a perfect square. Of all the options given, only 4 is the perfect square.

9(x-y) = 9*4 = (3^2)(2^2)
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VP
Joined: 07 Dec 2014
Posts: 1223
xy and yx are reversed two digit positive integers. If the difference  [#permalink]

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Updated on: 30 Sep 2017, 10:17
gracie wrote:
xy and yx are reversed two digit positive integers. If the difference between them is a perfect square>10, what is the difference between x and y?

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

another approach:
because the perfect square is the difference between reversed integers,
it must be a multiple of 9
the only perfect square multiples of 9 between 10 and 100 are 36 and 81
as 81 is too great to be the difference between reversed two digit integers,
then xy-yx=36➡
(10x+y)-(10y+x)=36➡
9x-9y=36➡
x-y=4
C

Originally posted by gracie on 30 Sep 2017, 09:58.
Last edited by gracie on 30 Sep 2017, 10:17, edited 1 time in total.
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Re: xy and yx are reversed two digit positive integers. If the difference  [#permalink]

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30 Sep 2017, 10:12
1
gracie wrote:
xy and yx are reversed two digit positive integers. If the difference between them is a perfect square>10, what is the difference between x and y?

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

There are no calculations needed to solve this question!

The difference of a 2 digit number and its reverse is $$10x+y - (10y+x) = 9(x-y)$$

From the answer options, only Option C(4) can be the difference of x and y,
as 9(x-y) is a perfect square only if x-y = 4.
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Re: xy and yx are reversed two digit positive integers. If the difference  [#permalink]

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01 Nov 2018, 10:34
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Re: xy and yx are reversed two digit positive integers. If the difference   [#permalink] 01 Nov 2018, 10:34
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