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

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

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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

Kudos [?]: 4 [1], given: 0

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S
Joined: 27 Dec 2016
Posts: 153

Kudos [?]: 38 [0], given: 192

Concentration: Social Entrepreneurship, Nonprofit
GPA: 3.65
WE: Sales (Consumer Products)
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Re: If x and y are positive integers, is x a multiple of y? [#permalink]

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New post 05 Sep 2017, 11:16
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. :sad: :sad:

Waiting for OE.
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Kudos [?]: 38 [0], given: 192

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

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New post 05 Sep 2017, 12:15
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 \(B\)

Kudos [?]: 175 [0], given: 31

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Kudos [?]: 38 [0], given: 192

Concentration: Social Entrepreneurship, Nonprofit
GPA: 3.65
WE: Sales (Consumer Products)
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If x and y are positive integers, is x a multiple of y? [#permalink]

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New post 05 Sep 2017, 12:40
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 :-D

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 8-)
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If x and y are positive integers, is x a multiple of y?   [#permalink] 05 Sep 2017, 12:40
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