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VP  P
Joined: 07 Dec 2014
Posts: 1221
xy and yx are a pair of reversed two digit positive integers. If x^2-y  [#permalink]

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4 00:00

Difficulty:   85% (hard)

Question Stats: 49% (02:34) correct 51% (02:05) wrong based on 86 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 V
Joined: 22 May 2016
Posts: 3670
xy and yx are a pair of reversed two digit positive integers. If x^2-y  [#permalink]

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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

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Director  P
Joined: 24 Nov 2016
Posts: 798
Location: United States
xy and yx are a pair of reversed two digit positive integers. If x^2-y  [#permalink]

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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

$$1≤(x,y)≤9…2≤(x+y)≤18$$
$$x^2-y^2=45…(x+y)(x-y)=45…factor.pairs(45)=(45,1);(15,3);(9,5)$$
$$(x+y)(x-y)=(45)(1)…(x+y)=45=invalid…(x+y≤18)$$
$$(x+y)(x-y)=(15)(3)…(x+y)=15…(x-y)=3…2x=18…x=9…y=6$$
$$(x+y)(x-y)=(9)(5)…(x+y)=9…(x-y)=5…2x=14…x=7…y=2$$
$$(xy,yx)=(96,69);(72,72)=2.pairs$$

SVP  P
Joined: 03 Jun 2019
Posts: 1849
Location: India
Re: xy and yx are a pair of reversed two digit positive integers. If x^2-y  [#permalink]

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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

Given: xy and yx are a pair of reversed two digit positive integers.

Asked: If x^2-y^2=45, how many such pairs are there?

(x-y)(x+y) = 45 = 9*5 = 15*3

Case 1:
x-y = 5
x+y =9
(x,y) = (7,2)

Case 2:
x-y = 3
x+y = 15
(x,y) = (9,6)

xy = {72,27,96,69}
Pairs = 2

IMO C
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E-mail : kinshook.chaturvedi@gmail.com Re: xy and yx are a pair of reversed two digit positive integers. If x^2-y   [#permalink] 19 Oct 2019, 08:15
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