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If XY is divisible by 4, which of the following must be true [#permalink]

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29 Nov 2008, 14:27

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If XY is divisible by 4, which of the following must be true?

A. If X is even then Y is odd B. If X = sqrt(2) then Y is not a positive integer C. If X is 0 then X + Y is not 0 D. X^Y is even E. X/Y is not an integer

If XY is divisible by 4, which of the following must be true?

1. If X is even then Y is odd 2. If X = sqrt(2) then Y is not a positive integer 3. If X is 0 then X + Y is not 0 4. X^Y is even 5. X/Y is not an integer

Do we assume that both X and Y can be the same number? Does this go the same in the actual GMAT Test?

None. Seems the question is flawed.

1: It is not true if x = 2 and y = 2. 2: It is not true if y = 2 sqrt(2). Y doesnot need to be -ve. 3: It is not true if y = 0. 4: It is not true if x = 4 and y = 3. 5: It is not true if x = 4 and y = 2.

Only choice B must be true. If X is sqrt(2) then Y cannot be an integer, otherwise it would mean that XY is not an integer. Counter-examples to other answer choices:

If \(XY\) is divisible by 4, which of the following must be true?

* If \(X\) is even then \(Y\) is odd. * If \(X = \sqrt{2}\) then \(Y\) is not a positive integer. * If \(X\) is 0 then \(X + Y\) is not 0. * \(X^Y\) is even. * \(\frac{X}{Y}\) is not an integer.

Can someone explain how the answer is B? I feel even if x = rt2, Y can be a positive integer.

If \(XY\) is divisible by 4, which of the following must be true?

* If \(X\) is even then \(Y\) is odd. * If \(X = \sqrt{2}\) then \(Y\) is not a positive integer. * If \(X\) is 0 then \(X + Y\) is not 0. * \(X^Y\) is even. * \(\frac{X}{Y}\) is not an integer.

Can someone explain how the answer is B? I feel even if x = rt2, Y can be a positive integer.

\(X = \sqrt{2}\) and Y = 4

XY = (rt2 * 4) /4 which is divisible by 4 = rt2

The confusing part is the word "positive": If \(x=\sqrt{2}\), \(y\) can be positive as well as negative BUT it can not be an integer! It can not be positive integer.

\(y\) must be of a type of \(y=\sqrt{2}*2n\) where \(n\) is an integer, not necessarily positive. So \(y\) can be \(\sqrt{2}*8\), when \(n=4\) OR \(-\sqrt{2}*2\), when \(n=-1\) OR \(0\), when \(n=0\).

As when \(y=\sqrt{2}*2n\) and \(x=\sqrt{2}\), \(xy=4n\) and the condition that it's divisible by \(4\) holds true for any value of \(n\).
_________________

Bunuel - I still didn't get it why Y " must be". Why can Y not be positive or why can Y not be an integer??

The question states whether XY is divisible by 4. It does not state that the result of XY must be an integer. So even if X is \(\sqrt{2}\) , Y can be a multiple of 4 and XY can be divisible by 4, resulting in \(\sqrt{2}\)

Bunuel - I still didn't get it why Y " must be". Why can Y not be positive or why can Y not be an integer??

The question states whether XY is divisible by 4. It does not state that the result of XY must be an integer. So even if X is \(\sqrt{2}\) , Y can be a multiple of 4 and XY can be divisible by 4, resulting in \(\sqrt{2}\)

In the following:

X = \(\sqrt{2}\) and Y = 4

XY = 4\(\sqrt{2}\)

XY/4 = \(\sqrt{2}\) 4/4

=\(\sqrt{2}\)

First of all XY is divisible by 4 means that the product of XY is evenly divided by 4, the result must be an integer. So \(x=\sqrt{2}\) and \(y=4\), will not work as the result \(\sqrt{2}\) is not an integer.

Second Y can be positive. Y can not be positive integer. Y must have \(\sqrt{2}\) in it to "compensate" \(\sqrt{2}\) of \(X\), to turn their product, \(XY\), into the integer, so that \(XY\) to have a chance of divisibility by \(4\). When \(Y\) has \(\sqrt{2}\) in it, multiplied by \(X=\sqrt{2}\), the product will be an integer.

So: Y can be negative irrational number: \(-\sqrt{2}*2\), \(-\sqrt{2}*4\), \(-\sqrt{2}*6\), \(-\sqrt{2}*8\), etc; OR Y can be positive irrational number: \(\sqrt{2}*2\), \(\sqrt{2}*4\), \(\sqrt{2}*6\), \(\sqrt{2}*8\), etc; OR Y can be zero, then XY=0 and 0 is divisible by any number, including 4.

Re: If XY is divisible by 4, which of the following must be true [#permalink]

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10 Nov 2012, 15:39

i think the thing that needs to be made clear is that whenever gmat says that a number is divisible by an integer then that implies that both the numerator and the denominator are integers and in our case XY product has to be integer.

i think the thing that needs to be made clear is that whenever gmat says that a number is divisible by an integer then that implies that both the numerator and the denominator are integers and in our case XY product has to be integer.

On GMAT when we are told that \(a\) is divisible by \(b\) (or which is the same: "\(a\) is multiple of \(b\)", or "\(b\) is a factor of \(a\)"), we can say that: 1. \(a\) is an integer; 2. \(b\) is an integer; 3. \(\frac{a}{b}=integer\).

So the terms "divisible", "multiple", "factor" ("divisor") are used only about integers (at least on GMAT).

BELOW IS REVISED VERSION OF THIS QUESTION:

If \(x\) and \(y\) are positive integer and \(xy\) is divisible by 4, which of the following must be true?

A. If \(x\) is even then \(y\) is odd. B. If \(x\) is odd then \(y\) is a multiple of 4. C. If \(x+y\) is odd then \(\frac{y}{x}\) is not an integer. D. If \(x+y\) is even then \(\frac{x}{y}\) is an integer. E. \(x^y\) is even.

Notice that the question asks which of the following MUST be true not COULD be true.

A. If \(x\) is even then \(y\) is odd --> not necessarily true, consider: \(x=y=2=even\);

B. If \(x\) is odd then \(y\) is a multiple of 4 --> always true: if \(x=odd\) then in order \(xy\) to be a multiple of 4 y mst be a multiple of 4;

C. If \(x+y\) is odd then \(\frac{y}{x}\) is not an integer --> not necessarily true, consider: \(x=1\) and \(y=4\);

D. If \(x+y\) is even then \(\frac{x}{y}\) is an integer --> not necessarily true, consider: \(x=2\) and \(y=4\);

E. \(x^y\) is even --> not necessarily true, consider: \(x=1\) and \(y=4\);

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