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Are x and y integers? (1) The product xy is an integer. (2) x + y

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Bunuel
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Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   03 Jan 2016, 13:52
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E

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33% (01:16) correct 67% (01:20) wrong based on 189 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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E
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chetan2u
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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   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..
hence answer is E
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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   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? (1) The product xy is an integer. (2) x + y [#permalink] New post   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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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   03 Jan 2016, 19:57
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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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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   26 Jul 2019, 05:18
Lets x = -square root 2
y = square root 2

(1) xy = -2. Insufficient

(2) x+y = 0. Insufficient

(1)+(2), Also gives no information regarding whether x and y are integers or not

E is correct
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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   10 Jun 2021, 07:49
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For the product xy to be an integer, x and y both necessarily may not be integers
eg: x=0.2 and y=10 gives xy=2

But for x+y to be an integer, both must necessarily be integers
eg: 1+0 = 1, 2+(-2)=0

Therefore, Statement (2) is sufficient.
Answer (B)
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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   10 Jun 2021, 07:58
1) If both are integers, we will get an integer. However, let x=0.5 and y=6 xy=3
INSUFFICIENT

2) Once again, if both are integers it will yield an integer. However, let x = 0.5 and y = 0.5 and then x+y = 1
INSUFFICIENT

1+2) Does not give us anything new.

Answer E
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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   10 Jun 2021, 08:07
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imo C

1+2= on combining provides....y(y+int) = int

so Y will be integer

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Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   Updated on: 14 Jun 2021, 09:54
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Stmt 1
2/3 * 3/2 = 1 (Integer)

hence insufficient.

Stmt 2

1/3+2/3 = 1 (Integer)

hence insufficient.

Combining both if we take x & y both as root2
We get yes & no.

Hence E is the answer

Originally posted by SangamJena on 10 Jun 2021, 10:07.
Last edited by SangamJena on 14 Jun 2021, 09:54, edited 1 time in total.
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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   12 Jun 2021, 05:42
I'm going with C.

When you combine 1 and 2, there is no way you can have fractions for x and y. So it will be C.
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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   12 Jun 2021, 07:03
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Bunuel wrote:
Are x and y integers?

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


This problem turned out to be more challenging than I though it would be. C is not correct. Oddly enough Princeton Review also says that the OA is C and is wrong about it. The correct answer is E.

If \(x=\sqrt{2}\) and \(y=-\sqrt{2}\) then the answer is NO but if \(x = y = 1\), the answer is YES. Not sufficient.

Answer: E.
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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink] New post   04 Apr 2023, 09:35
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Re: Are x and y integers? (1) The product xy is an integer. (2) x + y [#permalink]
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