Bunuel wrote:
Is the integer x odd?
(1) 2(y+ x) is an odd integer.
(2) 2y is an odd integer.
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:(1) INSUFFICIENT: 2(y + x) is an odd integer. How is it possible that 2 multiplied by something could yield an odd integer? The value in the parentheses must not be an integer itself. For example, the decimal 1.5 times 2 yields the odd integer 3. List some other possibilities:
2(y + x) = 1, 3, 5, 7, 9, etc.
(y + x) = 1/2, 3/2, 5/2, 7/2, 9/2, etc.
You know that x is an integer, so y must be a fraction in order to get such a fractional sum. Say that y = 1/2. In that case, x = 0, 1, 2, 3, 4, etc. Thus, x can be either odd (“yes”) or even (“no”).
(2) INSUFFICIENT: This statement tells you nothing about x. If 2y is an odd integer, this implies that y = odd/2 = 1/2, 3/2, 5/2, etc.
(1) AND (2) INSUFFICIENT: Statement (2) fails to eliminate the case you used in Statement (1) to determine that x can be either odd or even. Thus, you still cannot answer the question with a definite yes or no.
But, just to combine the statements another way,
Statement (1) says that 2(y + x) = 2y + 2x = an odd integer.
Statement (2) says that 2y = an odd integer. By substitution, odd + 2x= odd, so 2x = odd - odd = even.
2x would be even regardless of whether x is even or odd.
The correct answer is (E).
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