题目信息
If x and z are integers, is x + z2 odd?
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B:Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D:EACH statement ALONE is sufficient.
E:Statements (1) and (2) TOGETHER are NOT sufficient.
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已选答案：
正确答案：
D:EACH statement ALONE is sufficient.
Arithmetic Properties of integers
We are given that x is odd and z is even. Therefore, z2 is even and hence x + z2 is odd, because an odd integer added to an even integer is an odd integer; SUFFICIENT. We are given that x – z is odd. Since there is not a readily apparent useful algebraic relation between x – z and x + z2, we consider all possible cases.
From the table it is clear that if x – z is odd, then x + z2 is odd; SUFFICIENT.
The correct answer is D;each statement alone is sufficient.
We are given that x is odd and z is even. Therefore, z2 is even and hence x + z2 is odd, because an odd integer added to an even integer is an odd integer; SUFFICIENT. We are given that x – z is odd. Since there is not a readily apparent useful algebraic relation between x – z and x + z2, we consider all possible cases.
| x | z | z2 | x − z | x + z2 |
| even | even | even | even | even |
| even | odd | odd | odd | odd |
| odd | even | even | odd | odd |
| odd | odd | odd | even | even |
From the table it is clear that if x – z is odd, then x + z2 is odd; SUFFICIENT.
The correct answer is D;each statement alone is sufficient.
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