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# If x and y are positive integers, is 2x a multiple of y?

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If x and y are positive integers, is 2x a multiple of y? [#permalink]

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01 Jan 2006, 14:03
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If x and y are positive integers, is 2x a multiple of y?

(1) 2x + 2 is a multiple of y
(2) y is a multiple of x
[Reveal] Spoiler: OA

Last edited by Bunuel on 02 Nov 2013, 05:28, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: DS - Is 2x a multiple of y [#permalink]

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10 Aug 2011, 14:21
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This one isn't clear to me...Like to explain why it is E and not A

IF x and y are positive integers, is 2x a multiple of y?
a. 2x + 2 is a multiple of y
b. y is a multiple of x

From a:

If x = 2 and y = 3, (2x+2) is a multiple of y but not 2x.
If x = 2 and y = 2, (2x+2) and 2x are multiple of y.

So not sufficient.

From b: y is a multiple of x means x is equal to or smaller than y.

If y = x = 3, 2x is a multiple of y.
If y = 3 and x = 1, 2x is not a multiple of y. not sufficient.

From a and b: (2x + 2) is a multiple of y and y is a multiple of x.

If x = 2 and y = 2, (2x+2) is multiple of y and y is also a multiple of x. So 2x is also a multiple of y.
If y = 6 and x = 2, (2x+2) is multiple of y and y is also a multiple of x but 2x is not a multiple of y.

So together also not sufficient. E.
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Re: DS - Is 2x a multiple of y [#permalink]

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10 Aug 2011, 16:27
statement 1. just means that y is even, but it does not tell us anything about the direct relationship between x and y. Simplified is really saying 2x + 2 =yk , k being an unknown multiplication of y

as the earlier post did, plug numbers.

2(2) + 2 = yk : when x is 2, y can be 2 multiplied 3 times, and so 2x will be a multiple of y BUT we could also assume y is 3 multiplied 2 times, in which case 2x ie. 4 is not a multiple of 3. this gives a yes/no answer so its insufficient.

2(x) + 2 = y (k times)
2(2) + 2 = 2 (3 times) ...........4 is a multiple of 2
BUT
2(2) + 2 = 3 (2 times)...........4 is not a multiple of 3
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Re: DS - Is 2x a multiple of y [#permalink]

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13 Aug 2011, 16:10
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$$x, y > 0$$, integers

"Is $$\frac{2x}{y}$$ an integer?"
"Is $$2x$$ a multiple of (divisible by) $$y$$?"
"Is $$y$$ a factor of $$2x$$?"

Statement 1) This statements says that $$\frac{2x + 2}{y} = \frac{2(x+1)}{y}$$ is an integer. In other words, $$2(x+1)$$ is a multiple of $$y$$.

In order for both $$2x$$ and $$2(x+1)$$ to be divisible by $$y$$, $$y$$ must be either $$1$$ (factor of $$2$$ and both $$x$$ and $$(x+1)$$) or $$2$$ (factor of $$2$$). Thus if $$y$$ equals $$1$$ or $$2$$, then $$\frac{2x}{y}$$ is an integer. If $$y$$ has any other value under this constraint, $$\frac{2x}{y}$$ is not an integer.

You can also break this statement into two expressions: $$\frac{2x + 2}{y} = \frac{2x}{y} + \frac{2}{y}$$. If $$\frac{2x}{y} + \frac{2}{y}=integer$$, then in order for $$\frac{2x}{y}$$ to be an integer, $$\frac{2}{y}$$ must also be an integer. Thus if $$y$$ equals $$1$$ or $$2$$, then $$\frac{2x}{y}$$ is an integer. If $$y$$ has any other value, $$\frac{2x}{y}$$ is not an integer.

Insufficient.

Statement 2) This statement says that $$\frac{y}{x}$$ is an integer. So $$y = x*k$$, where $$k$$ is some integer.

In order for $$\frac{2x}{y} = \frac{2*x}{k*x}$$ to be an integer, $$\frac{2}{k}$$ must be an integer, ie $$k$$ must equal either $$1$$ or $$2$$. Thus if $$y = 2x$$ or $$y = x$$, $$2x$$ will be a multiple of $$y$$. With any other value of $$k$$, $$2x$$ will not be a multiple of $$y$$.

Insufficient.

Combined)
From statement 1, $$\frac{2x}{y}$$ is an integer when $$y={1,2}$$, and not an integer when $$y$$ has another value.
From statement 2: $$\frac{2x}{y}$$ is an integer when $$y=2x$$ or $$y=x$$, and not an integer when $$y$$ is a different multiple of $$x$$ than $$2$$.

So under the constraints of both statements, $$\frac{2x}{y}$$ is an integer when $$(x,y)$$ = $$(1,1)$$, $$(2,2)$$, or $$(1,2)$$.

It only takes one example that follows the two constraints but is not one of these 3 possibilities to prove the combined statements insufficient. A couple examples are $$(x,y) = (1,4)$$, or $$(2,6)$$.

[Reveal] Spoiler:
E
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Is 2x a multiple of y? [#permalink]

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30 May 2012, 13:38
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Is 2x a multiple of y?

(1) 2x+2 is a multiple of y
(2) y is a multiple of x
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Re: Is 2x a multiple of y? [#permalink]

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31 May 2012, 01:24
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kashishh wrote:
Is 2x a multiple of y?

(1) 2x+2 is a multiple of y
(2) y is a multiple of x

stmnt1:

If y= 1 x=2 then 2x+2 = 6 and multiple of y. 2x =4 and multiple of y
but if y=3 and x=2 then 2x+2=6 and multiple of y but 2x=4 and not a multiple of y. Hence insuff

stmnt 2:

y = 2 and x= 1 then 2x is multiple of y
y= 3 and x= 1 then 2x is not multiple of y. Hence insuff

taking together
x=1 and y =2 we have y is multiple of x, 2x+2 = 4 multiple of y and 2x= 2 is multiple of y (2) .
x=1 and y =4 we have y is multiple of x, 2x+2 = 4 multiple of y but 2x= 2 is not multiple of y (4)

Hence E
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Re: IF x and y are positive integers, is 2x a multiple of y? [#permalink]

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01 Nov 2013, 21:20
Bunuel,

w/o plugin is their a better way to solve this question.
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Re: IF x and y are positive integers, is 2x a multiple of y? [#permalink]

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02 Nov 2013, 05:32
honchos wrote:
Bunuel,

w/o plugin is their a better way to solve this question.

It's easy to get a NO and an YES answers with number plunging, so yes, this way is probably the fastest/easiest.
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Re: Is 2x a multiple of y? [#permalink]

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15 Jul 2014, 06:54
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Re: Is 2x a multiple of y? [#permalink]

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15 Jul 2014, 23:29
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kashishh wrote:
Is 2x a multiple of y?

(1) 2x+2 is a multiple of y
(2) y is a multiple of x

Use some logic and some number plugging:

Question: Is 2x a multiple of y?

(1) 2x+2 is a multiple of y

(2x + 2) is a multiple of y.
2x and (2x+2) are two consecutive even numbers and hence share only 2 factors: 1 and 2
So 2x will also be a multiple of y only if y is 1 or 2. Not sufficient.

(2) y is a multiple of x
Here are the multiples: x....y....2x
If y is x or 2x, 2x will be a multiple of y, else it will not be. Not sufficient.

Using both together:
If y = 2 and x = 2, all conditions are met and 2x IS a multiple of y.
If y = 6 and x = 2, all conditions are met but 2x IS NOT a multiple of y.
Not sufficient.

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Re: Is 2x a multiple of y? [#permalink]

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17 Jul 2014, 03:17
What if x=1 and y=2..it satisfies all the conditions??
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Re: Is 2x a multiple of y? [#permalink]

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17 Jul 2014, 03:26
Statment 1 tells that y is either 1 or 2.
Statment 2 shows that y is 2 as its a multiple of x and 1 cant be a multiple of any number, thats what understand, tell me where am going wrong!
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Re: Is 2x a multiple of y? [#permalink]

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17 Jul 2014, 06:03
What if x=1 and y=2..it satisfies all the conditions??

Statment 1 tells that y is either 1 or 2.
Statment 2 shows that y is 2 as its a multiple of x and 1 cant be a multiple of any number, thats what understand, tell me where am going wrong!

From (1), from 2x+2 is a multiple of y, we don't have that y is either 1 or 2. It can take many other values.

Is 2x a multiple of y?

(1) 2x+2 is a multiple of y. If x=1 and y=1, then the answer is YES but if x=1 and y=4, then the answer is NO. Not sufficient.

(2) y is a multiple of x. If x=1 and y=1, then then the answer is YES but if x=1 and y=4, then the answer is NO. Not sufficient.

(1)+(2) The same here: if x=1 and y=1, then then the answer is YES but if x=1 and y=4, then the answer is NO. Not sufficient.

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Re: If x and y are positive integers, is 2x a multiple of y? [#permalink]

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15 Nov 2015, 09:17
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Re: If x and y are positive integers, is 2x a multiple of y? [#permalink]

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22 Nov 2016, 12:23
Hello from the GMAT Club BumpBot!

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Re: If x and y are positive integers, is 2x a multiple of y?   [#permalink] 22 Nov 2016, 12:23
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