If 9 is a factor of 2x, then which of the following may not

Question

If 9 is a factor of 2x, then which of the following may not be an integer?

A. 6x/54 + 2x/3

B. (4x - 18)/9

C. (2x + 27)/9

D. (81 - 4x^2)/81

E. (2x - 3)/3

gayathri · Accepted Answer

NM, the question says "which of the following   may not   be an integer?"

May Not is the key word in the question.
Lets look at choice A:
6x/54 + 2x/3 = (3/6)(2x/9) + 2x/3 = (1/2)(2x/9) + 2x/3

So, as long as 2x/9 is a multiple of two it will be an integer. In your case you are using 2(18)/9 = 4, a multiple of two and hence the answer is an integer. 

It does not say x is an integer, so if you pick x=13.5 (2x=27 is divisible by 9), we get 
6x/54 + 2x/3 = 1.5 + 9 = 10.5, not an integer.

Questor · Answer

I got the right answer A after plugging in 18 for x.

While going through Kaplan's explanation, I couldn't understand the part about choice A.

"6x/54 + 2x/3 = 3/6 * 2x/9 + 2x/3.

Since 9 is a factor of 2x, 3 is also factor of 2x. So 2x/3 is an integer, as is 2x/9. But 3/6 = 1/2, so if 2x/9 is not even, the expression 3/6 * 2x/9 will not be even."

But, isn't 2x/9 always even? 9 is a factor of 2x, which means that we could have 2(9), 2(2)(9), 2(3)(9), and so forth. The 9 in the denominator cancels out the 9 in the numerator. So we are left with 2 * something. So 2x/9 is even. 

Your thoughts?

NM · Answer

The question tells us that 9 is a factor of 2x. 

Therefore, 2x = 9k where k is any integer.

2x = 9k. Therefore 6x = 27k. 

But 6 is an even number. Therefore k also has to be an even number. But if k necessarily has to be an even number, then 6x must be divisible by 54.

Therefore, we can say that 6x = 54m for some integer m....... (1)

Since 2x is divisible by 9, it is also divisible by 3........... (2)

By (1) and (2), we can see that 6x/54 + 2x/3 is an integer. 

So, the answer is  NOT  choice A.

gayathri · Answer

A.

Stem: 2x/9 = n where n is any integer.

a. 6x/54 + 2x/3 (lets look at other choices)

b. (4x-18)/9 = 4x/9 -18/9 = 2.2x/9 - 2 = 2n -2 = integer

c. (2x+27)/9 = 2x/9 + 27/9 = n+3 = integer

d. (81-4x^2)/81 = 81/81 - (2x/9)^2 = 1-n^2 = integer

e. (2x-3)/3 = 2x/3 - 3/3 = 2x/3 - 1 = integer (since 2x is divisible by 9, it is also divisible by 3)

The only choice that's left is A.

NM · Answer

Questor,

If you plug in 18 for x, then - 

(6 * 18)/54 + (2 * 18)/3 = 2 + 12 = 14 (which is an integer).

As my earlier post showed, choice A  cannot  be the correct answer.

patrickpui · Answer

my choice is D. the tricky part of the question is that D is written as (81-4x^2)/81 . With the (x-y), we must solve the (81-4x^2) before doing the division with 81. The result is often a fraction.

rxs0005 · Answer

All 

the answer to this is simple since 9 is a factor of 2x that means x can be and integer then its a multiple of 9 itself then choices A - E will hold good and we will always get an integer. 

BUT the problem does not say that x is an intger so take x = 4.5 so 2x will be 9 and 9 is a factor of 9 plug in the choices A - E you will only get D as an decimal value not an integer 

ANSWER IS D 

rxs0005

Questor · Answer

Hi. Thanks for the explanations, guys! I'll check the CD again to make sure that I didn't make any typos in my original.

gayathri · Answer

rxs0005, I dont see how you get D. 

Lets use x=4.5 as you have suggested:
a. 6x/54 + 2x/3 = 6(4.5)/54 +9/3 = 3.5 (NOT an integer)

b. (4x-18)/9 = (4(4.5)-18)/9 = (18-18)/9 = 0 (an integer)

c. (2x+27)/9 = (9+27)/9 = 4 (an integer)

d. (81-4x^2)/81 = (81 - 4(4.5)^2)/81 = (81-81)/81 = 0 (an integer) 

e. (2x-3)/3 = (9-3)/3 = 2 (an integer)

FN · Answer

RX you are wrong. the answer is A, you have to think of in terms of prime facrtos

2X/9 is like 2X/3.3 right?

so lets look at answer choice A

6x/54 is not an integer!

6X = 2.3.X right?
54= 3.3.2.3

6x/54 = 2.3.x/3.3.2.3 as you can see, 2.3 in the numerator cancels out with the ones in the denominator, right? you are left with X/3.3 which is never stated to be an integer, remember 2x/3.3 is an integer not x/3.3!

twixt · Answer

A it is.

What you learn is that 2x = 9N where N is equal to or greater than 1.

if 2x = 9, then A is not an integer, for BCDE 2x=9 yields to integer results...

ggurface · Answer

If 9 is a factor of 2x => 2x = 9k, where k = 0, 1, etc

6x/54 + 2x/3 can be rewritten in terms of k as:
27k/54 + 9k/3

which is k/2 + 3k

the 3k will always be an integer, but the first term only works for even k.

alternatively, if you just plugged in k=1 => 2x=9 => x=4.5 you would have arrived at A aswell

kham71 · Answer

The answer is A.

The questions provides that 2x is divisible by 9, so x is either a multiple of 9 or x = 9/2 = 4.5

If x = 4.5 then 6x/54 = (6*4.5)/ (6*9) = 1/2. Not an integer.

In all other cases when the equations are simplified, 2x is always the numerator and 9 or 3 always the denominator making the answer always an integer.

mikeyg51 · Answer

Thank you very much, I wrongly assumed (and i should be beat for it because I knew better!) that x is an integer...I should've picked a more uncommon value for x in which 2x/9 still holds true. Again, thank you

EMPOWERgmatRichC · Answer

Hi All,

Tougher GMAT questions are sometimes designed to test the thoroughness of your thinking. This prompt NEVER stated that X was an integer; it even goes so far as to state that 9 is a factor of 2X (not that 9 is a factor of X). When a prompt is built around some quirky language or 'design elements', it's often worthwhile to ask yourself what you really KNOW and what you don't know. As has been pointed out in several of the explanations, thinking about X=4.5 is the easiest way to get to the correct answer (even though 4.5 isn't the easiest value that fits the given description).

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

kingbucky · Answer

Why are you assuming x is an integer ?

Posted from my mobile device

terminatork07 · Answer

9 is a factor of 2x. So 9 CAN BE EQUAL to 2x -> x = 4.5

How put 4.5 or 9/2 as x in answer choices.

Only A gives non-integer answer.

saikrishna96 · Answer

Your reasoning is almost valid. However, if k=1, which is also possible, then 6x/54 will be 27/54, thus a non integer. So A may not be an integer in this case

Kinshook · Answer

If 9 is a factor of 2x, then which of the following may not be an integer?

9 is a factor of 2x.

2x = 9y; x = 4.5y : where y is an integer

A. 6x/54 + 2x/3 = 6*4.5y/54 + 2*9y/3 = .5y + 6y = 6.5y: MAY OR MAY NOT BE AN INTEGER

B. (4x - 18)/9 = (4*4.5y - 18)/9 = 2y - 2: INTEGER

C. (2x + 27)/9 = (2*4.5y + 27)/9 = y + 3: INTEGER

D. (81 - 4x^2)/81 = (81 - 4(9y/2)^2)/81 = 1 - y^2: INTEGER

E. (2x - 3)/3 = (2*4.5y - 3)/3 = 3y - 1 : INTEGER

IMO A

If 9 is a factor of 2x, then which of the following may not

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If 9 is a factor of 2x, then which of the following may not

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