请教一道GWD数学

If x + y is an integer, is y an integer?

(1)     x – y is an integer.

(2)     x + 2y is an integer.

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.

我的想法是这样的：

x+y= 某一整数 A

x-y=某一整数 B

把上式相加，则2x=某两整数之和，则x一定为整数，同样可得 y也是整数

所以我选 D

请多指教啊，我也只是推理