题目信息
If x and y are integers, is xy + 1 divisible by 3 ?
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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正确答案:
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Arithmetic Properties of integers
Determine whether xy + 1 is divisible by 3, where x and y are integers.
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 + 1)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. 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 + 1)(9r + 8) + 1 = 27qr + 9r + 24q + 9 = 3(9qr + 3r + 8q + 3), which is divisible by 3.
The correct answer is C;both statements together are sufficient.
Determine whether xy + 1 is divisible by 3, where x and y are integers.
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 + 1)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. 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 + 1)(9r + 8) + 1 = 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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