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
OA E
Source: Veritas Prep
For integers x and y, when x is divided by y, the remainder
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Since, the question asks "Which of the following must be true?", we can eliminate any answer choice that is not necessarily true.BTGmoderatorDC wrote: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
OA E
Source: Veritas Prep
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
Cheers,
Brent
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.
Regards!
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.
Regards!
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If both x and y are even, then the remainder when x is divided by y must be even (including 0). So at least one of x and y is odd.BTGmoderatorDC wrote: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
OA E
Source: Veritas Prep
Answer: E
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