If x is a positive number, is x an even integer?

[Vincen]

If x is a positive number, is x an even integer?

(1) 3x is an even integer.

(2) 5x is an even integer.

The OA is C.

Are both conditions necessary? I think (2) is the same as (1), isn't it?

[Jay@ManhattanReview]

If x is a positive number, is x an even integer?

(1) 3x is an even integer.
(2) 5x is an even integer.

The OA is C.

Are both conditions necessary? I think (2) is the same as (1), isn't it?

Yes, both the statements are fundamentally the same, but none by itself is sufficient.

Statement 1: 3x is an even integer.

Case 1: Say 3x = 6, then x = 2, even.
Case 2: Say 3x = 2, then x = 2/3, not an integer. Insufficient.

Statement 2: 5x is an even integer.

Case 1: Say 5x = 10, then x = 2, even.
Case 2: Say 5x = 2, then x = 2/5, not an integer. Insufficient.

Statement 1& 2:

Say 3x = 2a and 5x = 2b, whether a and b are positive integers.

Thus, 3/a = 5/b

=> 3b = 5a

Since a and b are positive integers, a must be a multiple of 3 and b must be a multiple of 5.

Say a = 3p, whether p is a positive integer.

Thus, x = 2a/3 = (2*3p) / 3 = 2p --> Integer. Sufficient.

The correct answer: C

Hope this helps!

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-Jay
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Hi Vincen,

We're told that X is a POSITIVE number (meaning that X > 0). We're asked if X is an EVEN INTEGER. This is a YES/NO question. This question can be solved by TESTing VALUES.

1) 3X is an even integer.

IF....
X = 2/3, then 3X = 2 and the answer to the question is NO.
X = 2, then 3X = 6 and the answer to the question is YES.
Fact 1 is INSUFFICIENT

2) 5X is an even integer.
X = 2/5, then 5X = 2 and the answer to the question is NO.
X = 2, then 5X = 10 and the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, the various 'fractional' answers that fit one of the Facts will NOT fit the other (since multiplying the same fraction by 3 and by 5 won't 'cancel out' the denominator and lead to an integer result for both calculations). Thus, the only possible values for X must be INTEGERS. In addition, since 3 and 5 are both ODD integers, for the 3X and 5X to both be EVEN integers, the X MUST be EVEN.
Combined, SUFFICIENT

Final Answer: C

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