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A
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If x and y are integers, is y an even integer?
(1) 4y^2+3x^2=x^4+y^4
(2) y=4−x^2
M27-02
(1) 4y^2+3x^2=x^4+y^4
(2) y=4−x^2
The official answer is A, and the logic is clear to me.
But, is it possible in the first equations also to have x = y = o? Shouldn't the answer be E in such case?
It is not explicitly stated in the wording that x and y are different non-zero integers?
Most probably I just missed smth, so would be grateful for your explanations
Thanks in advance!
But, is it possible in the first equations also to have x = y = o? Shouldn't the answer be E in such case?
It is not explicitly stated in the wording that x and y are different non-zero integers?
Most probably I just missed smth, so would be grateful for your explanations
Thanks in advance!
M27-02
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05 Jul 2014, 05:14
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Maksym
If x and y are integers, is y an even integer?
(1) 4y^2+3x^2=x^4+y^4 --> rearrange: \(3x^2-x^4=y^4-4y^2\) --> \(x^2(3-x^2)=y^2(y^2-4)\). Notice that LHS is even for any value of \(x\): if \(x\) is odd then \(3-x^2=odd-odd=even\) and if \(x\) is even then the product is naturally even. So, \(y^2(y^2-4)\) is also even, but in order it to be even \(y\) must be even, since if \(y\) is odd then \(y^2(y^2-4)=odd*(odd-even)=odd*odd=odd\). Sufficient.
(2) y=4-x^2 --> if \(x=odd\) then \(y=even-odd=odd\) but if \(x=even\) then \(y=even-even=even\). Not sufficient.
Answer: A.
As for your doubt: 0 is an even integer. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.
Check more tips on zero and number properties here: tips-and-hints-for-specific-quant-topics-with-examples-172096.html#p1371030
This week's PS question
This week's DS Question
Theory on Number Properties: math-number-theory-88376.html
DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59
This week's DS Question
Theory on Number Properties: math-number-theory-88376.html
DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59
Hope it helps.
General Discussion
05 Jul 2014, 03:40
Kudos
Bookmarks
Maksym,
Even if x= y= 0, then too Y will remain an even integer because 0 is an even integer. and as this is a question of data interpretation in which we need to look for UNIQUE SOLUTION and in both the cases i.e.
1)when y is not equal to zero
2) WHEN X=Y=0
we are getting unique answer in both the cases i.e. YES Y is an even integer.
so answer would be A instead of E.
Even if x= y= 0, then too Y will remain an even integer because 0 is an even integer. and as this is a question of data interpretation in which we need to look for UNIQUE SOLUTION and in both the cases i.e.
1)when y is not equal to zero
2) WHEN X=Y=0
we are getting unique answer in both the cases i.e. YES Y is an even integer.
so answer would be A instead of E.
