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# If x and y are positive integers and x^2 + y^2 = 100

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If x and y are positive integers and x^2 + y^2 = 100 [#permalink]  18 Aug 2012, 14:22
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Difficulty:

35% (medium)

Question Stats:

63% (01:34) correct 38% (01:17) wrong based on 8 sessions
If x and y are positive integers and $$x^2 + y^2 = 100$$, then the greatest possible value of x is between
(A) 4 and 5
(B) 6 and 7
(C) 7 and 8
(D) 9 and 10
(E) 10 and 11

I don't agree with the OA. According to the question, x is an integer, therefore the greatest value of x could be 8 (Choice C).
However, the OE is this:
To maximize x, minimize y. Since both variables are positive integers, the smallest y could be is 1. Thus:
$$x^2 + y^2 = 100$$
$$x^2 + 1^2 = 100$$
$$x^2 = 99$$
Since 100 is $$10^2$$, 99 must be the square of a little less than 10. The correct choice is (D).

Source: http://www.gmathacks.com
[Reveal] Spoiler: OA

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Re: If x and y are positive integers and x^2 + y^2 = 100 [#permalink]  18 Aug 2012, 14:37
metallicafan wrote:
If x and y are positive integers and $$x^2 + y^2 = 100$$, then the greatest possible value of x is between
(A) 4 and 5
(B) 6 and 7
(C) 7 and 8
(D) 9 and 10
(E) 10 and 11

I don't agree with the OA. According to the question, x is an integer, therefore the greatest value of x could be 8 (Choice C).
However, the OE is this:
To maximize x, minimize y. Since both variables are positive integers, the smallest y could be is 1. Thus:
$$x^2 + y^2 = 100$$
$$x^2 + 1^2 = 100$$
$$x^2 = 99$$
Since 100 is $$10^2$$, 99 must be the square of a little less than 10. The correct choice is (D).

Source: http://www.gmathacks.com

Both x and y can take integer values between 1 and 9, so not really sure as to the phrasing of the question, especially the word between.
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Re: If x and y are positive integers and x^2 + y^2 = 100 [#permalink]  18 Aug 2012, 21:31
I am confused as well:

The statement is:

$$x^2 + y^2 = 100$$

To make this add to 100 using positive integers with the OA would be to use non integer numbers:

The square root of 99 is not an integer for one, and 0 is not allowed to be used given the parameters in the question.

The only answer I got was C.

Where x=8 and y=6

8^2+6^2=100

I think that this is a flawed question.
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Re: If x and y are positive integers and x^2 + y^2 = 100 [#permalink]  19 Aug 2012, 00:54
If x is an integer, then without even reading the rest of the question none of the answer choices could possibly be right ('between' means 'strictly between' unless you add the word 'inclusive').

I don't know if there's a typo in the original question - I guess so, considering how the OE reads - but the question makes no sense as written. I suppose it makes sense if you take out the word 'integers' and just require x and y to be positive.

edit: or perhaps the question means to say "If $$x^2$$ and $$y^2$$ are positive integers..." Then the solution is right.
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Re: If x and y are positive integers and x^2 + y^2 = 100   [#permalink] 19 Aug 2012, 00:54
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