题目信息
If b is the product of three consecutive positive integers c, c + 1, and c + 2, is b a multiple of 24 ?
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：
A:Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Arithmetic Properties of integers
Since 24 = 23 × 3, and 1 is the only common factor of 2 and 3, any positive integer that is a multiple of 24 must be a multiple of both 23 = 8 and 3. Furthermore, the product of any three consecutive positive integers is a multiple of 3. This can be shown as follows. In b = c(c + 1)(c + 2), when the positive integer c is divided by 3, the remainder must be 0, 1, or 2. If the remainder is 0, then c itself is a multiple of 3. If the remainder is 1, then c = 3q + 1 for some positive integer q and c + 2 = 3q + 3 = 3(q + 1) is a multiple of 3. If the remainder is 2, then c = 3r + 2 for some positive integer r and c + 1 = 3r + 3 = 3(r + 1) is a multiple of 3. In all cases, b = c(c + 1)(c + 2) is a multiple of 3.
It is given that b is a multiple of 8. It was shown above that b is a multiple of 3, so b is a multiple of 24; SUFFICIENT. It is given that c is odd. If c = 3, then b = (3)(4)(5) = 60, which is not a multiple of 24. If c = 7, then b = (7)(8)(9) = (24)(7)(3), which is a multiple of 24; NOT sufficient.
The correct answer is A;statement 1 alone is sufficient.
Since 24 = 23 × 3, and 1 is the only common factor of 2 and 3, any positive integer that is a multiple of 24 must be a multiple of both 23 = 8 and 3. Furthermore, the product of any three consecutive positive integers is a multiple of 3. This can be shown as follows. In b = c(c + 1)(c + 2), when the positive integer c is divided by 3, the remainder must be 0, 1, or 2. If the remainder is 0, then c itself is a multiple of 3. If the remainder is 1, then c = 3q + 1 for some positive integer q and c + 2 = 3q + 3 = 3(q + 1) is a multiple of 3. If the remainder is 2, then c = 3r + 2 for some positive integer r and c + 1 = 3r + 3 = 3(r + 1) is a multiple of 3. In all cases, b = c(c + 1)(c + 2) is a multiple of 3.
It is given that b is a multiple of 8. It was shown above that b is a multiple of 3, so b is a multiple of 24; SUFFICIENT. It is given that c is odd. If c = 3, then b = (3)(4)(5) = 60, which is not a multiple of 24. If c = 7, then b = (7)(8)(9) = (24)(7)(3), which is a multiple of 24; NOT sufficient.
The correct answer is A;statement 1 alone is sufficient.
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