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If x and y are integers greater than 1, is x a multiple of y

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If x and y are integers greater than 1, is x a multiple of y [#permalink] New post 29 Jan 2012, 15:21
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If x and y are integers greater than 1, is x a multiple of y?

(1) 3y^2+7y=x
(2) x^2-x is a multiple of y


[Reveal] Spoiler:
For me its D. Unless someone else thinks otherwise. This is how I solved this:

Statement 1

y is a multiple of 3 and 7

So when y = 2 then x will be 26 and YES x is a multiple of y.
when y = 5 then x will be 110 and YES x is a multiple of y.

Therefore, statement 1 is sufficient to answer this questions.

Statement 2

x^2-x is a multiple of y

x(x-1) is a multiple of y. Sufficient to answer.

Therefore D for me i.e. both statements alone are sufficient to answer this question. Any thoughts guys?
[Reveal] Spoiler: OA

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Last edited by Bunuel on 30 Oct 2012, 00:44, edited 2 times in total.
Added the OA
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Re: Is x a multiple of y [#permalink] New post 29 Jan 2012, 15:31
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If x and y are integers great than 1, is x a multiple of y?

(1) 3y^2+7y=x --> y(3y+7)=x --> as 3y+7=integer, then y*integer=x --> x is a multiple of y. Sufficient.

(2) x^2-x is a multiple of y --> x(x-1) is a multiple of y --> x can be multiple of y (x=2 and y=2) OR x-1 can be multiple of y (x=3 and y=2) or their product can be multiple of y (x=3 and y=6). Not sufficient.

Answer: A.

Hope it helps.
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Re: If x and y are integers greater than 1, is x a multiple of y [#permalink] New post 18 Jun 2012, 06:07
Hi Bunuel,
Can you please explain this problem. I did not understand the explanation above for Point 2.
Sorry for the inconvenience.
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Re: If x and y are integers greater than 1, is x a multiple of y [#permalink] New post 18 Jun 2012, 06:10
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harshavmrg wrote:
Hi Bunuel,
Can you please explain this problem. I did not understand the explanation above for Point 2.
Sorry for the inconvenience.


Can you please be a little bit more specific? What exactly didn't you understand in the second statement?
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Re: If x and y are integers greater than 1, is x a multiple of y [#permalink] New post 05 Sep 2012, 07:00
Hi Bunuel,
I have not understood the answer to the (2) statement, infact we have that x(x-1) = Y*F (because is a multiple), doesn't it depend on the number F if X is a multiple of y? For example:
If x=2, then y could be either 2 or 1; so the answer would be yes;
If x=3, then y could be either 2 or 3; so the answer would be no;
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Re: If x and y are integers greater than 1, is x a multiple of y [#permalink] New post 05 Sep 2012, 13:17
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for(2)

x(x-1) is multiple of y.
so either x is divisible by y or x-1 is divisible by y.
if x-1 is divisible by y, then x is not divisble by y.

plug in y=8, x=16--->16*15 is divisible by 8 --> yes x is a multiple of y.
plug in y=8,x=17----> 17*16 is still divisible by 8 --> But x is not a multiple of y.

So insufficient.
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Is x multiple of y? If: 3y^2 + 7y = x? If: x^2-x is mult? [#permalink] New post 31 Dec 2012, 16:16
x and y are integers > 1. Is x multiple of y?

(1): 3y^2 + 7y = x
(2): x^2 - x is multiple of y
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Re: Is x multiple of y? If: 3y^2 + 7y = x? If: x^2-x is mult? [#permalink] New post 31 Dec 2012, 20:27
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Nadezda wrote:
x and y are integers > 1. Is x multiple of y?

(1): 3y^2 + 7y = x
(2): x^2 - x is multiple of y


The question is asking whether x/y is an integer or not.
Answer is A.

Statement 1 can be written as y(3y +7)=x or 3y+7=x/y. Since y is an integer, therefore 3y+7 must also be an integer. Hence x/y will be an integer or x is a multiple of y. Sufficient.

Statement 2 can be written as x(x-1)/y is an integer. Notice that we can't deduce that x is a multiple of y because it is quite possible that the product is a multiple of y, but not the individual entities.
ex. 2*3/6 is a mltiple of 6 BUT neither 2 nor 3 is a multiple of 6.
Insufficient.

+1A

Please do add the OA while posting questions.
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Re: Is x multiple of y? If: 3y^2 + 7y = x? If: x^2-x is mult? [#permalink] New post 31 Dec 2012, 21:41
Nadezda wrote:
x and y are integers > 1. Is x multiple of y?

(1): 3y^2 + 7y = x
(2): x^2 - x is multiple of y


Hi Nadezda,

The Questions asks whether x/y is an integer or not.

from St 1 we have 3y^2 +7y=x ---> x/y= 3y+7 if y=2, x=13 and 13/2 is not an integer.
If y=7, then x/y=4 which is an integer.
So St1 alone not sufficient

From St 2 we have x^2-x is multiple of y ----> x(x-1)/y= Integer value
Now we know x-1 and x are consecutive integers

X2-x is even because if x is odd then x2-x (odd-odd)=even and if x is even then x2-x is even. Now x can be odd or even and hence alone not sufficient.

Combining both statement,we get for x/y to be an integer y has to be 7 and x will be have to be multiple of 7.
But we are not sure from above statement so ans should be E.

What is OA??

Thanks
Mridul
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Re: Is x multiple of y? If: 3y^2 + 7y = x? If: x^2-x is mult? [#permalink] New post 31 Dec 2012, 21:59
Expert's post
mridulparashar1 wrote:
Nadezda wrote:
x and y are integers > 1. Is x multiple of y?

(1): 3y^2 + 7y = x
(2): x^2 - x is multiple of y


Hi Nadezda,

The Questions asks whether x/y is an integer or not.

from St 1 we have 3y^2 +7y=x ---> x/y= 3y+7 if y=2, x=13 and 13/2 is not an integer.
If y=7, then x/y=4 which is an integer.
So St1 alone not sufficient

From St 2 we have x^2-x is multiple of y ----> x(x-1)/y= Integer value
Now we know x-1 and x are consecutive integers

X2-x is even because if x is odd then x2-x (odd-odd)=even and if x is even then x2-x is even. Now x can be odd or even and hence alone not sufficient.

Combining both statement,we get for x/y to be an integer y has to be 7 and x will be have to be multiple of 7.
But we are not sure from above statement so ans should be E.

What is OA??

Thanks
Mridul


if x/y=3y+7, then taking y as 2 will yield x/y as 13 and not 13/2.
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Re: Is x multiple of y? If: 3y^2 + 7y = x? If: x^2-x is mult? [#permalink] New post 01 Jan 2013, 02:07
Expert's post
Nadezda wrote:
x and y are integers > 1. Is x multiple of y?

(1): 3y^2 + 7y = x
(2): x^2 - x is multiple of y


Merging similar topics. Please refer to the solutions above.

Also, please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to the rules #5 and 7.
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Re: If x and y are integers greater than 1, is x a multiple of y [#permalink] New post 03 May 2014, 11:26

FACTS ESTABLISHED BY STEMS


  • x is an integer
  • y is an integer
  • x > 1
  • y > 1

QUESTION TRANSLATED


Since both x and y are integers the question "is x a multiple of y?" really asks if

x = q \cdot y + r

where q is an integer (the quotient) and r is the remainder when you divide x by y so that r = 0 which means that

x = q \cdot y + 0

which implies that

x = q \cdot y

So the question translates to:

is x = q \cdot y when x, q and y are all integers, and when x>1 and y > 1?

STATEMENT 1



3y^2 + 7y = x

can be rewritten as

y(3y + 7) = x

however, from the question stems we know that y is an integer, which implies 3y is an integer and that 3y+7 is also an integer:

y \cdot integer = x

which can be rewritten as

x = y \cdot integer or better yet, as

x = integer \cdot y

which if you notice is an equation of the form x = q \cdot y since q is an integer which implies that x is indeed a multiple of y.

Therefore, Statement A is sufficient.

STATEMENT 2


the statement "x^2 - x is a multiple of y" must be rewritten as "x (x - 1) is a multiple of y" which implies that

either x is a multiple of y or x-1 is a multiple of y

the sub-statement x is a multiple of y implies that x = q \cdot y which would answer the question, but we still need to figure out if the second sub-statement answers the question with the same answer as the first sub-statement.

the second sub-statement x-1 is a multiple of y implies that

x-1 = q \cdot y

which can be rewritten as

x = q \cdot y + 1

which is an equation of the form x = q \cdot y + r when r=1 which implies that when x is divided by y we get a remainder of 1, meaning that x is NOT a multiple of y.

Since both sub-statements are contradictory, this means that when x^2 - x is a multiple of y, x is not always a multiple of y. meaning that we cannot determine whether x is always a multiple of y according to Statement 2 alone.

Therefore, Statement 2 is not sufficient.

ANSWER



(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Re: If x and y are integers greater than 1, is x a multiple of y   [#permalink] 03 May 2014, 11:26
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