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# Are x and y integers?

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Math Expert
Joined: 02 Sep 2009
Posts: 50039
Are x and y integers?  [#permalink]

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03 Jan 2016, 13:52
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Question Stats:

38% (01:28) correct 62% (01:12) wrong based on 83 sessions

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Are x and y integers?

(1) The product xy is an integer.
(2) x + y is an integer.

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Math Expert
Joined: 02 Aug 2009
Posts: 6976
Re: Are x and y integers?  [#permalink]

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03 Jan 2016, 19:55
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BrainLab wrote:
Bunuel wrote:
Are x and y integers?

(1) The product xy is an integer.
(2) x + y is an integer.

(1) 2*2 YES, 2*1/2 NO
(2) 2+2 YES, 0.5 + 0.5 NO
(1)+(2) hmm... for statement 1 to be an integer we need either 2 integers or reciprocals, but I don't think that there are reciprocals that total an integer, that's why only integer pair satisfies both expressions. Please correct me if I'm wrong...

Hi,
in these type of Q one should know that there are various complex numbers which can fit in..
say $$x=2+\sqrt{3}$$ and $$y=2-\sqrt{3}$$
so xy=(2+\sqrt{3})*(2-\sqrt{3})=1..
and x+y= 2+\sqrt{3}+2-\sqrt{3}=4..

here the reason is we can take these two as roots of an equation, then
the eq becomes..
z^2+(x+y)z+xy... this will always give you the roots x and y as int ,frac or complex number, even if the product and sum of roots are integer..
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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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Re: Are x and y integers?  [#permalink]

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03 Jan 2016, 17:03
Bunuel wrote:
Are x and y integers?

(1) The product xy is an integer.
(2) x + y is an integer.

(1) 2*2 YES, 2*1/2 NO
(2) 2+2 YES, 0.5 + 0.5 NO
(1)+(2) hmm... for statement 1 to be an integer we need either 2 integers or reciprocals, but I don't think that there are reciprocals that total an integer, that's why only integer pair satisfies both expressions. Please correct me if I'm wrong...
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Re: Are x and y integers?  [#permalink]

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03 Jan 2016, 19:10
I tried the following

For statement 1, x = 2 and y = 1/2, x*y = 2*1/2 = 1 (an integer) No
x= 2 and y = 1, x*y = 2*1 = 2(also an integer) Yes

For statement 2, x = 2.5,y=1.5, x+y = 2.5+1.5 (an integer) No
x= 2, y = 1, x+y = 2+1 = 3( an integer) Yes

Combining 1+2, we might arrive at an answer E, how ever consider the possibility that x = 0 and y = 0 , both are integers
and this would satisfy both the conditions and so the answer is C according to me.
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Posts: 6976
Re: Are x and y integers?  [#permalink]

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03 Jan 2016, 19:57
Pragnya87 wrote:
I tried the following

For statement 1, x = 2 and y = 1/2, x*y = 2*1/2 = 1 (an integer) No
x= 2 and y = 1, x*y = 2*1 = 2(also an integer) Yes

For statement 2, x = 2.5,y=1.5, x+y = 2.5+1.5 (an integer) No
x= 2, y = 1, x+y = 2+1 = 3( an integer) Yes

Combining 1+2, we might arrive at an answer E, how ever consider the possibility that x = 0 and y = 0 , both are integers
and this would satisfy both the conditions and so the answer is C according to me.

Hi,
you can always find values of x and y as an integers satisfying the given conditions..
BUT we have to be sure that there are no values fittin as non integer, ecause the Q is a 'must ' Q..
hope it helped
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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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Re: Are x and y integers?  [#permalink]

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09 Jun 2018, 12:17
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Re: Are x and y integers? &nbs [#permalink] 09 Jun 2018, 12:17
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