2012-04-25 09:00:53

If 3xy is an odd integer, and both x and y are integers, which of the following MUST be true?

(A) x is not odd
(B) x/y is not odd
(C) y is not odd
(D) x/y is not even
(E) x/y is not an integer

The correct answer is D

Explanation: The product of two odd numbers is odd, while the product of an even number and an odd or even number is even. Thus, if 3xy is odd, then xy must be odd (since 3 is odd, so only the product of 3 and an odd number would be odd). If xy is odd, then both x and y must be odd. This eliminates choices A and C, since "x or y is not odd" means "x or y is even." The quotient of two odd numbers is a number that is not even (though it could be a fraction or an integer), making choice D the best answer.

This could also be found by plugging in numbers. Let x = 3 and y = 5. Then 3xy would be 3(3)(5), or 45. Use these numbers in the answer choices to see what works, remembering that "must be true" means that there cannot be a single number that works for 3xy but does not work for the answer choices:

(A) x is not odd: No, 3 is odd.
(B) x/y is not odd: This holds true for 3 and 5, since 3/5 is not odd (nor even).
(C) y is not odd: No, 5 is odd.
(D) x/y is not even: This holds true for 3 and 5, since 3/5 is not even (nor odd).
(E) x/y is not an integer: This holds true for 3 and 5, since 3/5 is not an integer.

Now, try a second set of numbers. Let x =9 and y = 3. Then, 3xy would be 3(9)(3), or 81. Use these numbers in the remaining choices:

(B) x/y is not odd: No, 9/3 is 3, which is odd.
(D) x/y is not even: Yes, 9/3 is 3, which is odd (and, thus, not even).
(E) x/y is not an integer: No, 9/3 is 3, which is an integer.