2. 外语
  3. If x and y are integers, is xy + 1 divisible by 3 ? (1) When x is divided by 3, the remainder is 1. (2) When y is divided by 9,

If x and y are integers, is xy + 1 divisible by 3 ? (1) When x is divided by 3, the remainder is 1. (2) When y is divided by 9,

admin2022-10-18  55
问题 If x and y are integers, is xy + 1 divisible by 3 ?
(1) When x is divided by 3, the remainder is 1.
(2) When y is divided by 9, the remainder is 8.
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 xy+ 1 is divisible by 3, where x and y are integers.
(1)     It is given that the remainder is 1 when x is divided by 3. It follows that x = 3q + 1 for some integer q. So, xy + 1 = (3q + l)y + 1. If y = 2, then xy + 1 = 6q + 3, which is divisible by 3. However, if y = 1, then xy + 1 = 3q + 2, which is not divisible by 3; NOT sufficient.
(2)     It is given that the remainder is 8 when y is divided by 9. It follows that y = 9r + 8 for some integer r. So, xy + 1 = (9r + 8)x + 1. If x = 1, then xy+1 = 9r+9, which is divisible by 3. However, if x = 2, then xy + 1 = 18r + 17, which is not divisible by 3; NOT sufficient.
Taking (1) and (2) together gives x = 3q + 1 and y = 9r + 8, from which it follows that xy + 1 = (3q+l)(9r+8) + l = 27qr+9r+24q + 9 = 3(9qr + 3r +8q + 3), which is divisible by 3.
The correct answer is C;
both statements together are sufficient.
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