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  3. If b is the product of three consecutive positive integers c, c + 1, and c + 2,

If b is the product of three consecutive positive integers c, c + 1, and c + 2,

游客2025-02-27  12
问题 If b is the product of three consecutive positive integers c, c + 1, and c + 2, is b a multiple of 24?
(1) b is a multiple of 8.
(2) c is odd.
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 A
解析 Since 24 = 23 x 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.
(1)     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.
(2)     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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