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## is x an even integer?

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PGMAT Rising GMAT Star
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is x an even integer? Sat Jun 09, 2012 2:03 pm
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If x is a positive number, is x an even integer?

(1) 3x is an even integer.

(2) 5x is an even integer.

Can someone explain how to solve this?

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eagleeye GMAT Destroyer!
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Sat Jun 09, 2012 3:15 pm
Hi PGMAT:

The correct answer should be C. Let me explain:
Remember that x is a positive number, not necessarily an integer.

1) 3x is an even integer. Let 3x=2m (m is a positive integer). Here m=3/2*x
If x=2/3, then 3*2/3=2 but x is not an integer.
However if x=2, 3*2=6 is still an even ineger.
Not sufficient.
2) 5x is an even integer. Let 5x=2n (n is a positive integer). Here n=5/2*x
by the same logic, x can be 2/5 or 2, which still makes it insufficient.

EDIT: I had posted a harder to understand approach earlier, but I saw Mitch's post below and based the following explanation on that. I couldn't delete this post (the option dosen't come up for some reason), so I edited below, please thank Mitch if you like this approach as all credit goes to him)

Now combining the two, 2*3x-5x =6x-5x = x.
Since 6x, 5x are both even and we know that
even - even = even
Hence x is an even integer. Hence C is correct. Thank you Mitch!!

Let me know if this helps

Last edited by eagleeye on Sat Jun 09, 2012 5:42 pm; edited 4 times in total

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Sat Jun 09, 2012 4:12 pm
PGMAT wrote:
If x is a positive number, is x an even integer?

(1) 3x is an even integer.

(2) 5x is an even integer.

Can someone explain how to solve this?
Statement 1: 3x is even.
It's possible that x = 2, which is an even integer.
It's possible that x = 2/3, which is not an even integer.
INSUFFICIENT.

Statement 2: 5x is even.
It's possible that x = 2, which is an even integer.
It's possible that x = 2/5, which is not an even integer.
INSUFFICIENT.

Statements 1 and 2 combined:
5x-3x = even - even = even.
Since 5x-3x = 2x, 2x is even.
3x-2x = even - even = even.
Since 3x-2x=x, x is even.
SUFFICIENT.

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Thanked by: eagleeye, PGMAT
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IknowIcan Just gettin' started!
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Sat Jun 09, 2012 7:36 pm
Hi guys,

Can you elaborate this part:

3x-2x = even - even = even.
Since 3x-2x=x, x is even.
SUFFICIENT.

Also, how can we prove C by using the sample numbers rather than numbers properties or algebra?

Thanks!

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Sat Jun 09, 2012 8:32 pm
IknowIcan wrote:
Hi guys,

Can you elaborate this part:

3x-2x = even - even = even.
Since 3x-2x=x, x is even.
SUFFICIENT.

Also, how can we prove C by using the sample numbers rather than numbers properties or algebra?

Thanks!
When one even number is subtracted from another, the result is another even number.
Thus:
When even number 3x is subtracted from even number 5x, the result -- 2x -- is an another even number.
When even number 2x is subtracted from even number 3x, the result -- x -- is another even number.
Thus, when the two statements are combined, we know that x is even.
SUFFICIENT.

If we stick to plugging in values:
Non-integer values that satisfy statement 1 are 2/3, 4/3, 8/3, etc.
None of these values satisfies statement 2.
Non-integer values that satisfy statement 2 are 2/5, 4/5, 6/5, 8/5, etc.
None of these values satisfies statement 1.
Thus, the only values that satisfy BOTH statements are EVEN INTEGERS:
2, 4, 6, 8, etc.
Thus, to satisfy both statements, x must be even.
SUFFICIENT.

Generally, plugging in numbers is the easiest way to prove that a statement is INSUFFICIENT.
But to prove that a statement is SUFFICIENT, algebra or some other sort of reasoning often is more efficient.

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dhonu121 GMAT Destroyer!
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Mon Jun 11, 2012 2:03 am
Quote:
If x is a positive number, is x an even integer?

(1) 3x is an even integer.

(2) 5x is an even integer.

Can someone explain how to solve this?
1. 3x is even.
so 3x=2 or x=3/2=1.5. Not even.
also 3x=4 or x = 4/3=1.33. Not Even.
also 3x=6 or x=2. Hence Even.
Thus for some cases its even for some its odd. Hence nothing can be said from this info.
Hence Insufficient.

2. 5x is even.
Same as above.
5x=2 or 4 or 6. x is not even.
5x=10 or 20 or 30. X is even.
Thus nothing can be said.
Hence Insufficient.

Combining 1 and 2.
3x is even and 5x is even.
If 3x=2 or 4 or 8, then from that value of x, 5x is not even.
If 3x=6,12,18 then from that value of x, 5x is also even.
Thus for 1 and 2 to be true, x has to be even.

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