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题库 / Prep2007E1-DS-203
What is the remainder when the positive integer x is divided by 6?
(1) When x is divided by 2, the remainder is 1; and when x is divided by 3, the remainder is 0.
(2) When x is divided by 12, the remainder is 3.
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.
解：
条件(1)：如果x除2余1且x除3余0，那么可以得到x的表达式必然为3+6k（k = 0, 1, 2...）。这个表达式很容易得到，首先找到最小的x，由于3除3余0，且3除2余1，所以此处x最小为3。找到最小的x后，根据高斯同余定理，由于2和3的最小公倍数是6，所以3+6k依然可以保证除3余0，且除2余1。由于6k必然可以被6整除，所以3+6k除以6必然余3。
条件(2)：x的表达式为3+12k（k = 0, 1, 2...）。由于12k一定可以被6整除，所以3+12k除6必然余3。
因此，两个条件均可以单独推理出答案。
代数进去就是
(1) When x is divided by 2, the remainder is 1 --> x is an odd number AND "when x is divided by 3, the remainder is 0" --> x is a multiple of 3 --> so, x is an odd multiple of 3: 3, 9, 15, 21, ... --> you can see a definite pattern here that any such number divided by 6 yields remainder of 3. Sufficient.
想问一下为什么单独看（1）是对的？
我是这样想的：
它只说明是个3的倍数并且是个奇数（后者就无意义了），相当于只有前面这一个条件——这个数是3的倍数
我跟你有同样的疑惑，后来我发现，条件一成立，余数就一定是3，如果x是9，那么被6除余数就是3，同理27，81等等，不知道我说的是不是对的
嗯嗯嗯 懂啦 蟹蟹亲~
解析已更新。
谢谢老师！
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