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Brilliant question!! My initial thought was sqrt2 is a valid remainder. Haha. Surprising how assumptions can kill you!! Although I chose B (a good guess!)
I plugged in x is sqrt2, then argued with myself that Y can be four and the statement will hold true.
So, takeaway from this is REMAINDERS HAVE TO BE INTEGERS!
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) \(\frac{X}{Y}\) is not an integer.
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 \(y=4\);
I kinda agree with the above post by rongali. Here are my 2 cents:-
"Y cannot be an integer" is easily proved wrong since y can be zero and therefore XY will be divisible by 4.
If Y cannot be a positive integer, then, considering that GMAT revolves around only real numbers, Y can be Zero, a negative Integer, a positive fraction or a negative fraction. If Y is zero, then XY is always divisible by 4. If Y is a negative integer, then XY will not be divisible by 4. If Y is a positive fraction, then XY could be or could not be divisible by 4, and likewise if Y is a negative Fraction, then XY could be or could not be divisible by 4.
But i think the catch of the question is that we cannot prove the parent statement wrong. So considering that XY IS DIVISIBLE by 4, we will have to ignore the possibilities mentioned above that do no lead to a proper divisibility by 4. Therefore, we are left with Y=0 OR Y=Negative Fraction, that produces XY divisible by 4 OR Y=Positive Fraction that produces proper divisibility. All in All, we realize that Y cannot be a positive integer because in that case XY will never be divisible by 4. Phewww, i'm exhausted.
I kinda agree with the above post by rongali. Here are my 2 cents:-
"Y cannot be an integer" is easily proved wrong since y can be zero and therefore XY will be divisible by 4.
If Y cannot be a positive integer, then, considering that GMAT revolves around only real numbers, Y can be Zero, a negative Integer, a positive fraction or a negative fraction. If Y is zero, then XY is always divisible by 4. If Y is a negative integer, then XY will not be divisible by 4. If Y is a positive fraction, then XY could be or could not be divisible by 4, and likewise if Y is a negative Fraction, then XY could be or could not be divisible by 4.
But i think the catch of the question is that we cannot prove the parent statement wrong. So considering that XY IS DIVISIBLE by 4, we will have to ignore the possibilities mentioned above that do no lead to a proper divisibility by 4. Therefore, we are left with Y=0 OR Y=Negative Fraction, that produces XY divisible by 4 OR Y=Positive Fraction that produces proper divisibility. All in All, we realize that Y cannot be a positive integer because in that case XY will never be divisible by 4. Phewww, i'm exhausted.
nice question Answer is B. the thing is, Y is not positive integer. because if X=\sqrt{2}, Y must be either zero or non integer. it can be 4\sqrt{2} which is not positive integer. for short, assume B states if X=\sqrt{2}, Y is not integer (except 0)