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

Question

If x is a positive number, is x an even integer?
 
(1) 3x is an even integer.
 
(2) 5x is an even integer.

Bunuel · Accepted Answer

Notice that we are not told that x is an integer.
 
(1) 3x is an even integer. x could be ANY even number or some fraction (for example 2/3), so this statement is NOT sufficient.

(2) 5x is an even integer. The same here, x could be ANY even number or some fraction (for example 2/5). Not sufficient.

(1)+(2) We have that 3x=even and 5x=even. Subtract one from another: 5x-3x=even-even --> 2x=even --> x=even/2=integer. Now, x=integer and 3x=even (from 1) means that x must be an even integer. Sufficient.

Answer: C.

EvaJager · Answer

Neither (1) nor (2) alone is sufficient.

(1) and (2) together:

From (1)  3x=2k,  where  k  is a positive integer.
From (2)  5x=2m,  where  m  is some positive integer. Necessarily  \frac{2k}{3}=\frac{2m}{5}  from which  5k=3m. 
 k  and  m  being integers, necessarily  k  must be a multiple of 3 (because 5 is not divisible by 3), so  k=3a  for some positive integer  a. 
It follows that  x=\frac{2k}{3}=2a  so  x  is even.

Sufficient.

Answer C

ziko · Answer

My answer to this question was D, both sufficient, however my assumption was that in GMAT  number  and  integer  are interchangible words, but as i see i was wrong. Bunuel could you please remind what word was interchangible with word  integer ?

Bunuel · Answer

I think you refer to Natural Numbers, which are non-negative (or positive) integers but GMAT doesn't use words "Natural Number" in their questions.

So, there is no interchangeable word for "integer" on the GMAT.

domfrancondumas · Answer

I am a little confused.
1) 3x is an even integer.
 Let x=4/3; then 3x=4. But in this case x is not an even integer. Hence INSUFFICIENT.

2) 5x is an even integer.
 Let x=4/5; then 5x=4. But in this case x is not an even integer. Hence INSUFFICIENT.

(1+2): 15x is an integer.
Let x=4/15; then 15x=4. But in this case x is not an even integer. Hence INSUFFICIENT.

Bunuel · Answer

Notice that x cannot be 4/15, because in this case 3x=12/15 which is NOT an even integer and 5x=20/15 which is also NOT an even integer, so in this case both statements are violated.

kooks123 · Answer

If it's given that "5 is a factor of x". Is it correct to assume that x is an integer? If yes, how and why? Please explain. Also explain if the answer to my question is No. Consider a case wherein x is 5/2. Isn't 5 a factor of x? Please address, thanks a lot in advance

Bunuel · Answer

On the GMAT when we are told that  a  is divisible by  b  (or which is the same: " a  is multiple of  b ", or " b  is a factor of  a "), we can say that: 
1.  a  is an integer;
2.  b  is an integer;
3.  \frac{a}{b}=integer .

Mo2men · Answer

x is positive Integer means we need to think only in positive fractions and integers.

(1) 3x is an even integer.

Let x= 2.....3x=6..........x is even integer

Let x=2/3....3x=2.........x is NOT even integer

Insufficient

(2) 5x is an even integer.

Let x= 2........5x=10..........x is even integer

Let x=2/5.......5x=2...........x is NOT even integer

Insufficient

Combining 1 +2 ,

There is not fraction could be multiplied simultaneously to 3 & 5 and give EVEN INTEGER. We left with that x=EVEN INTEGER to give make both 3x & 5x even integers.

As 3 & 5 are odd numbers then x needs to be EVEN Integer to make both statements Even integers.

Answer: C

CrackverbalGMAT · Answer

Hi Ellipse,

Whenever you combine two statements in a DS question, the best thing to do instead of plugging in values is to use one statement into the other or use a mathematical operation between the two statements. The mathematical operation(s) that you choose to perform must lead to your target question . In this case the target question is 'Is x an even integer'?

Statement 1 : 3x is an even integer

Here x can either be an even integer such as 2, 4, 6.... or it can be a fraction such as 2/3, 4/3..... So x can be an even integer or a fraction.  Insufficient .

Statement 2 : 5x is an even integer

Here again x can be an even integer such as 2, 4, 6.... or it can be a fraction such as 2/5, 4/5.... So x again can either be an even integer or a fraction.  Insufficient .

Now instead of recycling the values it makes sense to use a mathematical operation (or mathematical operations) between the two statements.

Let us multiply the first statement by 2, since 3x = even integer ; 3x * 2 = even integer * 2 ----->  6x = even integer

Statement 2 says that 5x is an even integer and by multiplying statement 1 by 2 we have 6x to be an even integer. Subtracting the two we get

6x - 5x = even integer - even integer  ----->  x = even integer .  Sufficient .

Answer : C

Nunuboy1994 · Answer

Statement 1:

3 (2/3) could be an even number- the questin says x is a positive number...not necessarily an integer

Insufficient

Statement 2

5(2/5) btw again here were are using a counterexample to establish that x does not necessarily have to be an integer
10/5 =2

Insufficient

Statement 1 and 2:

Statement 1 and Statement 2

Knowing from statement 1 and 2 that 3x and 5x are even integers we can use solve this question using algebra instead of imagining numbers and testing various cases- an even integer minus an even integer is always an even integer so

5x-3x=2x
5(4)-3(4)=2(4)

HolaMaven · Answer

Dear, Natural numbers are not non-negative numbers as 0 is a non-negative Integer but 0 is not natural number. Natural numbers are positive integers.

Bunuel · Answer

There are two definitions as mentioned in my post: https://en.wikipedia.org/wiki/Natural_number https://mathworld.wolfram.com/NaturalNumber.html https://www.mathopenref.com/counting-number.html https://www.wolframalpha.com/input/?i=natural+numbers

HolaMaven · Answer

Thank you for your enlightenment on the topic. Do the definitions differ according to the area of usage? Like in Number Theory we have Natural Numbers starting from 1 and in Set Theory or Computer Science Natural Numbers includes 0.

sahilvijay · Answer

1) 3x is even with x=2 and x= 2/3 , hence A,D ruled out
2) 5x is even with x= 2 and x= 2/5 B ruled out

3x = 2,4,6,8,10,12..... 
x = 2/3, 4/3, 6/3,8/3,.....

5x = 2,4,6,8,10....
x= 2/5,4/5,6/5,8/5,10/5

taking common from above series x=2,4,6....and so on...hence even.
 sufficient and C is answer

Bunuel · Answer

Yes, there are different definitions:
"The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... (OEIS A000027) or to the set of nonnegative integers 0, 1, 2, 3, ... (OEIS A001477; e.g., Bourbaki 1968, Halmos 1974). Regrettably, there seems to be no general agreement about whether to include 0 in the set of natural numbers. In fact, Ribenboim states "Let P be a set of natural numbers; whenever convenient, it may be assumed that 0 element P."

Luckily we don't have to worry about this at all. Not a single official GMAT question uses a term "natural number".

GMAT questions use: integer, positive integer, negative integer, non-negative integer.

sahilvijay · Answer

Bunuel 
good day Bunuel,

let in case if a ques come in gmat then whether i have to include 0 in natural number or not. as u mentioned above that gmat has not asked natural numbers specifically in its questions.
till today i was knowing that 0 is a whole number. and is not a natural number.

P.S. thanks

EMPOWERgmatRichC · Answer

Hi All,

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

GMAT assassins aren't born, they're made, 
Rich

Elite097 · Answer

avigutman  I understand this question however i have a doubt in combining the two statements. If we have to combine the two, then what we call as a fraction is only the fraction in its reduced form or any kind of fraction? For example : On combining the two if we say x=30/15, then can ay tht this is also a possible value of x and since its a fraction then in a way saying that x is both int or fraction, making c insufficient?

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

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

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