Is z an even integer? (1) z/2 is an even integer (2) 3z is an even

Question

Is z an even integer?

(1) z/2 is an even integer.
(2) 3z is an even integer.

subhashghosh · Accepted Answer

(1)
Z = 2 * even

=> z is even

Sufficient

(2) 3z = even

3 is odd

So z is even if z is an integer

But if z = 8/3

Not Sufficient

Answer - A

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION:

The wording of this question has a tendency to bias people towards integers. After all, the “opposite” of even is odd, and odd numbers are integers, too. However, the question does not state that z must be an integer in the first place, so  do not assume  that it is.

(1) SUFFICIENT: The fact that z/2 is an even integer implies that z = 2 × (an even integer), which much be an even integer. (In fact, according to statement (1), z must be divisible by 4).

(2) INSUFFICENT: The fact that 3z is an even integer implies that z = (an even integer)/3, which might not be an integer at all. For example, z could equal 2/3.

One way to avoid assuming is to invoke Principle #3: Work from Facts to Question. Statement (2) tells us that 3z = even integer = –2, 0, 2, 4, 6, 8, 10, etc. No even integers have been skipped over, nor have we allowed the question to suggest z values. That is how assumptions sneak in.

Next, we divide 3z by 3 to get z, so we divide the numbers on our list by 3: z = –2/3, 0, 2/3, 4/3, 2, 8/3, 10/3, etc. Only then do we check this list against our question and see that the answer is Maybe.

The correct answer is A.

If we had assumed that z must be an integer, we might have evaluated statement (2) with two cases:
3 × even = even, so z could be even.
3 × odd = odd, so z is definitely not odd.
We would have  incorrectly  concluded that Statement (2) was sufficient and therefore  incorrectly  selected answer (D).

Another common assumption is that a variable must be positive. Do not assume that any unknown is positive  unless it is stated as such  in the information given (or if the unknown counts physical things or measures some other positive-only quantity).

Nayimoni · Answer

The trap for this problem is that there is no integer constraints. Let us look at each statement

1) z/2 is even so z = 4k and k is integer so z is definitely even integer. Suff. Cross BCE

2) 3z is even, so 3z = 2k, k integer. Z could be equal to 2/3 and 3z is even but answer NO, and z could be equal to 2 and answer is A. Insuff. Cross off D

Answer is A

Bunuel · Answer

Similar question to practice: if-z-is-an-integer-is-z-even-162351.html

AliciaSierra · Answer

OA is wrong. It should be D. Please edit OA from A to D

Bunuel · Answer

OA is NOT wrong.

(2) 3z is even --> if z = 2, then it's even but if z = 2/3, then it's not.

chetan2u · Answer

1) statement 1 clearly states z to be a multiple of 4.. so suff
2) statement two does not prove z to be an even integer.. insuff
 ans A

ManojReddy · Answer

Is z an even integer?

(1) z/2 is an even integer.

For z/2 to be even, z must be a multiple of 4. Meaning z takes the following values {4,8,12,16,20 ...}
so z is even integer.

Sufficient 
(2) 3z is an even integer.

{ Odd * Even = Even}
For 3z to be even integer. z must be an even integer, Since 3 is odd.
So z is even integer.

Sufficient 
 Ans:D

-Manoj Reddy
 Please press +1 Kudos if this post is helpful

shriramvelamuri · Answer

it has to be D.

1. Option z/2 = Even integer. => Z= 2 * even Integer => Z will be even. (Even * Even = Even)

2. 3Z is an even interger= Z must be even number for the end result to be even number.

So answer D.

chetan2u · Answer

Hi ManojReddy and shriramvelamuri , Its very important that we check answer for every possiblity.. look at statement 2.. (2) 3z is an even integer. z will be an even integer if z is an integer. But is that mentioned anywhere.. NO.. what if z is a fraction say 2/3... 3z=2,an even integer.. but z is not an even integer, not even an integer.. B is not suff.. hope it is clear..

jimwild · Answer

Please tell me i'm getting more and more confused, i wan to know on the actual GMAT, on the test, if they say

z/2 is divisible by 2 => then automatically we assume z/2 as integer, then z is integer right?

dipanjan93 · Answer

For Statement 1 - What if we assume z is 2/3 or any other real number as such? Here the value for z/2 will now become 1/3
Then in this case the answer would have been Option E. Please explain this scenario as well.

Bunuel · Answer

If z = 2/3, then z/2 is NOT even as stated in the first statement, so z cannot be 2/3.

BrentGMATPrepNow · Answer

Target question :    Is z an even integer?

Aside: Integer n is even if we can express n as n = 2k for some integer k

Statement 1 : z/2 is an even integer.  
This means z/2 =2k for some integer k
Multiply both sides by 2 to get: z = 4k 
This tells us that z is a multiple of 4, which means  z is definitely even 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : 3z is an even integer. 
 If  we were told that  z is an integer , then statement 2 would be sufficient. However, we aren't told that z is an integer. 
With this in mind, consider these two possible cases: 
 Case a : z = 2. Works because 3z = 3(2) = 6, and 6 is even. In this case, the answer to the target question is  YES, z is even 
 Case b : z = 2/3. Works because 3z = 3(2/3) = 2, and 2 is even. In this case, the answer to the target question is  NO, z is not even 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

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Is z an even integer? (1) z/2 is an even integer (2) 3z is an even

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Is z an even integer? (1) z/2 is an even integer (2) 3z is an even

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