For integers x and y, when x is divided by y, the remainder is odd. Which of the following must be true?

[BTGModeratorVI]

For integers x and y, when x is divided by y, the remainder is odd. Which of the following must be true?

A. x is odd
B. xy is odd
C. x and y share no common factors other than 1
D. The sum x + y is odd
E. At least one of x and y is odd

Answer: E
Source: Veritas Prep

[Brent@GMATPrepNow]

For integers x and y, when x is divided by y, the remainder is odd. Which of the following must be true?

A. x is odd
B. xy is odd
C. x and y share no common factors other than 1
D. The sum x + y is odd
E. At least one of x and y is odd

Answer: E
Source: Veritas Prep

Since, the question asks "Which of the following must be true?", we can eliminate any answer choice that is not necessarily true.
So let's test some values that satisfy the given conditions

For integers x and y, when x is divided by y, the remainder is odd.
One possible case is that x = 9 and y = 6 (since 9 divided by 6 leaves remainder 3)
Check the answer choices . . .
ELIMINATE B, since xy = 54, which is EVEN
ELIMINATE C, since 9 and 6 have a common factor of 3


Another possible case is that x = 6 and y = 5 (since 6 divided by 5 leaves remainder 1)
Check the answer choices . . .
ELIMINATE A since x is EVEN


Another possible case is that x = 11 and y = 5 (since 11 divided by 5 leaves remainder 1)
Check the answer choices . . .
ELIMINATE D since x is x + y = 16, which is EVEN

By the process of elimination, the correct answer is E

[swerve]

For integers x and y, when x is divided by y, the remainder is odd. Which of the following must be true?

A. x is odd
B. xy is odd
C. x and y share no common factors other than 1
D. The sum x + y is odd
E. At least one of x and y is odd

Answer: E
Source: Veritas Prep

Let \(X=nY+R\) (\(n=\) some quotient, \(R=\)remainder)

or \(X=nY+2k+1\) (\(2k+1=\)odd remainder)

Now if \(Y\) is even;

\(X=nY+2k+1=\)even\(+\)even\(+1=\)odd

and if \(Y\) is odd;

\(X=\)(odd or even)\(+\)(even)\(+1\)
\(=\)(odd or even)

So, at least \(1\) of \(X\) and \(Y\) will always be Odd.