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

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Intern
Joined: 28 May 2016
Posts: 2
If x and y are positive integers, is x a multiple of y? [#permalink]

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05 Sep 2017, 09:40
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Question Stats:

56% (01:31) correct 44% (01:54) wrong based on 32 sessions

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

(1) y^2 + y is not a factor of x.
(2) x^3 + x is not a multiple of y.
[Reveal] Spoiler: OA
Manager
Joined: 27 Dec 2016
Posts: 233
Concentration: Social Entrepreneurship, Nonprofit
GPA: 3.65
WE: Sales (Consumer Products)
Re: If x and y are positive integers, is x a multiple of y? [#permalink]

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05 Sep 2017, 10:16
sahuamit91 wrote:
If x and y are positive integers, is x a multiple of y?

(1) y^2 + y is not a factor of x.
(2) x^3 + x is not a multiple of y.

Interesting question! Trying to solve twice and still not get the right answer.

Waiting for OE.
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PS Forum Moderator
Joined: 25 Feb 2013
Posts: 836
Location: India
GPA: 3.82
If x and y are positive integers, is x a multiple of y? [#permalink]

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05 Sep 2017, 11:15
sahuamit91 wrote:
If x and y are positive integers, is x a multiple of y?

(1) y^2 + y is not a factor of x.
(2) x^3 + x is not a multiple of y.

The re-framed question can be Is $$x = yk$$ (where $$k$$ is any constant)

Statement 1: this implies that $$y^2 + y = y(y+1)$$ does not divide $$x$$. But $$x$$ may or may not be a multiple of $$y$$.
If $$x$$ is a multiple of $$y$$ i.e let $$x= yk$$, then as per the statement $$\frac{x}{y(y+1)}$$ is not divisible, or $$\frac{yk}{y(y+1)}$$
or $$\frac{k}{(y+1)}$$ or $$k$$ is not divisible by $$(y+1)$$. But $$x$$ can be a multiple of $$y$$. Similarly $$x$$ may not be a multiple of $$y$$ altogether. Hence the statement is not sufficient

Statement 2: this implies $$x^3+x$$ or $$x(x^2+1)$$ is not a multiple of $$y$$. This statement is Sufficient to prove that $$x$$ is not multiple of $$y$$ because if $$x$$ had been a multiple of $$y$$ then let $$x=yk$$
so $$x(x^2+1) = yk({y^2k^2}+1) =$$ multiple of $$y$$, which is not true as per the statement. Hence Sufficient

Option $$B$$
Manager
Joined: 27 Dec 2016
Posts: 233
Concentration: Social Entrepreneurship, Nonprofit
GPA: 3.65
WE: Sales (Consumer Products)
If x and y are positive integers, is x a multiple of y? [#permalink]

### Show Tags

05 Sep 2017, 11:40
sahuamit91 wrote:
If x and y are positive integers, is x a multiple of y?

(1) y^2 + y is not a factor of x.
(2) x^3 + x is not a multiple of y.

Got some enlighment here

Question ask whether $$\frac{x}{y}=integer$$.

Statement 1
- We can write the statement into $$\frac{x}{(y^2)+y}=non integer$$
- Simplify equation, we get $$\frac{x}{y(y+1)}=non integer$$ --> we isolate $$\frac{x}{y}$$ --> $$\frac{x}{y}=non integer*(y+1)$$
- From here, we can conclude that $$\frac{x}{y}$$ can be either an integer or not an integer.
- Case 1 : let say we have non integer $$\frac{1}{3}$$ and $$y=2$$, so $$\frac{1}{3}*(2+1) = integer$$.
- Case 2 : let say we have non integer $$\frac{1}{3}$$ and $$y=3$$, so $$\frac{1}{3}*(3+1) = non integer$$.
- INSUFFICIENT.

Statement 2
- We can write the statement into $$\frac{x^3+x}{y}=non integer$$
- Simplify equation, we get $$\frac{x(x^2+1)}{y}=non integer$$ --> we isolate $$\frac{x}{y}$$ --> $$\frac{x}{y}=\frac{non integer}{(x^2+1)}$$.
- From here, we can conclude that the result of $$\frac{x}{y}$$ always non integer. There is no way a non integer when divided by integer become integer.
- Thus, we got the definite answer here!
- SUFFICIENT

B
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If x and y are positive integers, is x a multiple of y?   [#permalink] 05 Sep 2017, 11:40
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# If x and y are positive integers, is x a multiple of y?

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