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

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Manager
Joined: 30 Oct 2004
Posts: 132
If 9 is a factor of 2x, then which of the following may not [#permalink]

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25 Dec 2004, 13:56
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9
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Question Stats:

51% (02:56) correct 49% (01:37) wrong based on 401 sessions

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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

The answer and my question are posted below.
[Reveal] Spoiler: OA
Manager
Joined: 30 Oct 2004
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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25 Dec 2004, 14:06
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.

Intern
Joined: 19 Aug 2004
Posts: 47
Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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25 Dec 2004, 17:25
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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.
Director
Joined: 07 Nov 2004
Posts: 683
Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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25 Dec 2004, 19:57
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.
Intern
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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26 Dec 2004, 15:21
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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.
Manager
Joined: 02 Dec 2004
Posts: 224
Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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27 Dec 2004, 11:18
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.
Director
Joined: 07 Nov 2004
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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28 Dec 2004, 17:11
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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.
Director
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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28 Dec 2004, 18:22
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

rxs0005
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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28 Dec 2004, 22:56
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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.
Director
Joined: 07 Nov 2004
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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31 Dec 2004, 18:17
rxs0005 wrote:
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

rxs0005

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)
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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01 Jan 2005, 11:58
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!

rxs0005 wrote:
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

rxs0005
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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01 Jan 2005, 15:10
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...
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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15 Sep 2014, 23:13
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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11 Aug 2015, 13:21
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
Intern
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If 9 is a factor of 2x, then which of the following may not [#permalink]

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29 Dec 2015, 05:43

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.
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Re: If 9 is a factor of 2x, then which of the following may not [#permalink]

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09 Oct 2016, 10:14
gayathri wrote:
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.

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
Re: If 9 is a factor of 2x, then which of the following may not   [#permalink] 09 Oct 2016, 10:14
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