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# xy and yx are a pair of reversed two digit positive integers. If x^2-y

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VP
Joined: 07 Dec 2014
Posts: 1118
xy and yx are a pair of reversed two digit positive integers. If x^2-y  [#permalink]

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26 Aug 2018, 14:03
4
00:00

Difficulty:

85% (hard)

Question Stats:

44% (02:27) correct 56% (02:11) wrong based on 43 sessions

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xy and yx are a pair of reversed two digit positive integers. If x^2-y^2=45, how many such pairs are there?

A. 0
B. 1
C. 2
D. 3
E. 4
Senior SC Moderator
Joined: 22 May 2016
Posts: 2117
xy and yx are a pair of reversed two digit positive integers. If x^2-y  [#permalink]

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26 Aug 2018, 17:44
1
gracie wrote:
xy and yx are a pair of reversed two digit positive integers. If x^2-y^2=45, how many such pairs are there?

A. 0
B. 1
C. 2
D. 3
E. 4

If $$x^2-y^2=45$$, then
$$(x+y)(x-y)=45$$

$$(x+y)=a$$: one factor of 45
$$(x-y)=b$$: another factor of 45

Factors of 45
$$(a)*(b)=45$$
$$1*45=45$$
$$3*15=45$$
$$5*9=45$$

The first set of factors will not work
$$x+y=45$$
$$x-y=1$$ Add
$$2x=46$$
$$x=23$$

Second set of factors
$$x+y=15$$
$$x-y=3$$ Add
$$2x=18$$
$$x=9$$

$$x-y=3$$
$$9-y=3$$
$$y=6$$

One pair, xy and yx: 96 and 69
$$(x^2-y^2)=(9^2-6^2)=(81-36)=45$$

Third set of factors
$$x+y=9$$
$$x-y=5$$ Add
$$2x=14$$
$$x=7$$

$$x-y=5$$
$$7-y=5$$
$$y=2$$

Another pair: 72 and 27
$$(x^2-y^2)=(7^2-2^2)=(49-4)=45$$

There are no more factor sets.

xy and yx are
96 and 69
72 and 27

Number of pairs: 2

xy and yx are a pair of reversed two digit positive integers. If x^2-y &nbs [#permalink] 26 Aug 2018, 17:44
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