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题库 / GMATAdvanced-DS-64
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.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
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x=3m+1; y=9k+8=3*3k+8, y/3=3*3k/3+8/3, 可知前面部分是可以除尽的，余数在8/3部分是2. x除以3的余数是1，y/3的余数是2，(xy+1)/3的余数=x/3的余数*y/3的余数+1/3的余数=1*2+1=3，所以余数是0.
求余时分母的加减法和乘法可以直接拆开；
因为xy+1能否被3整除，不但取决于x的值，而且还取决于y的值，所以(1)和(2)单独都很明显不充分；根据(1)+(2)可得：
x=3m+1，y=9n+8，其中m，n是整数。
把这两个式子代入xy+1可得：
xy+1=(3m+1)(9n+8)+1
=27mn+24m+9n+84-1
=3(9mn4-8m+3n+3)
因此xy+1能被3整除，所以(1)+(2)充分。