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  3. If x, y, and d are integers and d is odd, are both x and y divisible by d ?(1)

If x, y, and d are integers and d is odd, are both x and y divisible by d ?(1)

游客2025-02-27  23
问题 If x, y, and d are integers and d is odd, are both x and y divisible by d ?
(1) x + y is divisible by d.
(2) x - y is divisible by d.
选项 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.
答案 C
解析 Determine whether both of the integers x and y are divisible by the odd integer d.
(1)     It is given that x + y is divisible by d. It is possible that both x and y are divisible by d, and it is possible that they are not both divisible by d. For example, if x = 4, y = 2, and d= 3, then 4 + 2 is divisible by 3, but neither 4 nor 2 is divisible by 3. On the other hand, if x = 3, y = 6, and d=3, then 3 + 6 is divisible by 3, and both 3 and 6 are divisible by 3; NOT sufficient.
(2)     It is given that x - y is divisible by d. It is possible that both x and y are divisible by d, and it is possible that they are not both divisible by d. For example, if x = 4, y = -2, and d= 3, then 4 - (-2) is divisible by 3, but neither 4 nor -2 is divisible by 3. On the other hand, if x = 3, y = -6, and d= 3, then 3 - (-6) is divisible by 3, and both 3 and -6 are divisible by 3;
NOT sufficient.
Taking (1) and (2) together, x + y is divisible by d, so is an integer and x - y is divisible by d, so is an integer. It follows that is an integer and x/d is an integer
because d is odd. Similarly, is an integer and y/d is an integer because d is odd.
The correct answer is C;
both statements together are sufficient.
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