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# If x and y are distinct prime numbers, each greater than 2,

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If x and y are distinct prime numbers, each greater than 2, [#permalink]  13 Nov 2010, 10:09
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If x and y are distinct prime numbers, each greater than 2, which of the following must be true?

(I) x+y is divisible by 4
(II) x*y has even number of factors
(III) x+y has an even number of factors

A. I only
B. II only
C. I and III only
D. II and III only
E. I and II only

I plugged in numbers for each option, but it took me more than 2 mins to get to the right answer -OA is B.

I had to consider x=13 and y=23 for ruling out (III), since x+y in this case =36 which has odd no. of factors.

PLease explain if there is any other method for solving this under 2 mins

Thanks
[Reveal] Spoiler: OA
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If x and y are distinct prime numbers, each greater than 2, [#permalink]  13 Nov 2010, 10:48
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Q. If x and y are distinct prime numbers, each greater than 2, which of the following must be true?

(I) x+y is divisible by 4
(II)x * y has even number of factors
(III)x+y has an even number of factors

A. I only
B. II only
C. I and III only
D. II and III only
E. I and II only

I plugged in numbers for each option, but it took me more than 2 mins to get to the right answer -OA is B.

I had to consider x=13 and y=23 for ruling out (III), since x+y in this case =36 which has odd no. of factors.

PLease explain if there is any other method for solving this under 2 mins

Thanks

First of all we are asked "which of the following MUST be true" not COULD be true.

"MUST BE TRUE" questions:
These questions ask which of the following MUST be true, or which of the following is ALWAYS true no matter what set of numbers you choose. Generally for such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

As for "COULD BE TRUE" questions:
The questions asking which of the following COULD be true are different: if you can prove that a statement is true for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.

I. x+y is divisible by 4 --> if $$x=3$$ and $$y=7$$ then $$x+y=10$$, which is not divisible by 4. So this statement is not always true;

II. xy has even number of factors --> only perfect squares have an odd number of factors (check this: a-perfect-square-79108.html?hilit=perfect%20square%20reverse#p791479), as $$x$$ and $$y$$ are distinct prime numbers then $$xy$$ cannot be a perfect square and thus cannot have an odd number of factors, so $$xy$$ must have an even number of factors. This statement is always true;

III. x+y has an even number of factors --> now, $$x+y$$ can be a perfect square, for example if $$x=3$$ and $$y=13$$ then $$x+y=16=perfect \ square$$, so $$x+y$$ can have an odd number of factors. So this statement is not always true;

Hope it's clear.
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Re: Must be true - number properties [#permalink]  13 Nov 2010, 11:51
Yes, Thank you very much. You rock!
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Re: Must be true - number properties [#permalink]  13 Nov 2010, 12:10
Bunuel,

Does that mean that me never count 1 as a factor?
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Re: Must be true - number properties [#permalink]  13 Nov 2010, 12:17
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medanova wrote:
Bunuel,

Does that mean that me never count 1 as a factor?

No. 1 is a factor of every positive integer. Also 1 is a perfect square (1=1^2) and it has 1 (so odd) factor - itself.
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Re: If x and y are distinct prime numbers, each greater than 2, [#permalink]  29 Jan 2015, 12:21
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Re: If x and y are distinct prime numbers, each greater than 2, [#permalink]  29 Jan 2015, 12:38
Expert's post
Hi All,

Roman Numeral questions on the GMAT often come with a "design shortcut", meaning that you probably WON'T have to think about all 3 Roman Numerals (IF you pay attention to how the answer choices are designed).

Here, we're told that X and Y are DISTINCT (meaning "different") PRIME numbers that are each greater than 2. We're asked which of the following Roman Numerals MUST be true (which really means "which of these is ALWAYS TRUE no matter how many different examples you can come up with). It's often easier to try to prove that something is NOT true, so that should be our goal here - try to come up with a way to disprove a Roman Numeral so that we can ELIMINATE it.

TESTing VALUES will work perfectly here.

I. (X+Y) is divisible by 4

3+5 = 8 which is divisible by 4
3+7 = 10 which is NOT divisible by 4
Roman Numeral 1 is NOT always true, so we can ELIMINATE IT.

**NOTE: Look at the answers. Since Roman Numeral 1 has been eliminated, we can....
1) Eliminate answers A, C and E.
2) Notice that the two remaining answers both have Roman Numeral II in them, so that Roman Numeral MUST be TRUE. As such, we don't even have to deal with it!!!!!

III. (X+Y) has an even number of factors

3+5 = 8 which has 4 factors
3+7 = 10 which has 4 factors
3+11 = 14 which has 4 factors
3+13 = 16 which has 5 factors
Roman Numeral 3 is NOT always true, so we can ELIMINATE IT.

[Reveal] Spoiler:
B

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Re: If x and y are distinct prime numbers, each greater than 2,   [#permalink] 29 Jan 2015, 12:38
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