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# If x and y are non-zero integers, is x/y an integer?

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Math Expert
Joined: 02 Sep 2009
Posts: 52253
If x and y are non-zero integers, is x/y an integer?  [#permalink]

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26 Sep 2018, 04:25
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Difficulty:

75% (hard)

Question Stats:

57% (01:46) correct 43% (02:31) wrong based on 58 sessions

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If x and y are non-zero integers, is x/y an integer?

(1) $$\frac{x}{(y + 1)(y - 1)} = 1$$

(2) x - y = 2

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If x and y are non-zero integers, is x/y an integer?  [#permalink]

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26 Sep 2018, 12:55
Bunuel wrote:
If x and y are non-zero integers, is x/y an integer?

(1) $$\frac{x}{(y + 1)(y - 1)} = 1$$

(2) x - y = 2

As all we're given are equations, we'll work with them.
This is a Precise approach.

(1) Simplifying gives x = y^2 - 1 so x/y = (y^2 - 1)/y = y^2/y - 1/y = y - 1/y. This is an integer only if y = 1 which is impossilbe as then (y-1) would be 0 and (1) would have 0 in its denominator.
So y cannot be 1 and x/y is not an integer.
Sufficient.

(2) As this is one equation with 2 variables, it has infinite solutions and can be YES or NO depending on the specific values of x and y chosen. (for example x = 4, y = 2 gives yes and x = 5, y = 3 gives no)
Insufficient.

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Re: If x and y are non-zero integers, is x/y an integer?  [#permalink]

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29 Sep 2018, 23:11
Bunuel wrote:
If x and y are non-zero integers, is x/y an integer?

(1) $$\frac{x}{(y + 1)(y - 1)} = 1$$

(2) x - y = 2

X/Y can only be an integer if,
X/Y= odd/odd= odd
X/Y= even/odd= even
X/Y= even/even= Odd
X/Y= odd/even= not an integer

Let’s see S1,
X/ (Y+1) (Y-1) = 1
This is only possible when X is Odd and Y Even: O/ (E+O)*(E-O) = O/O*O = O/O= O
Which means, X/Y = O/E= not an integer (Sufficient)

Let’s see S2,
X-Y=2
This is only possible when X and Y are both Odd OR Even (O-O= E; E-E= E)
If both Odd: X/Y= O/O= O
If both Even: X/Y= E/E= O
This is not a unique answer, so it is not sufficient!

Ans is A.
Re: If x and y are non-zero integers, is x/y an integer? &nbs [#permalink] 29 Sep 2018, 23:11
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