AMC12 2018 A

AMC12 2018 A · Q12

AMC12 2018 A · Q12. It mainly tests Casework, Divisibility & factors.

Let $S$ be a set of 6 integers taken from $\{1, 2, \dots, 12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a < b$, then $b$ is not a multiple of $a$. What is the least possible value of an element of $S$?

设$S$是从$\{1, 2, \dots, 12\}$中取的6个整数的集合，具有如下性质：如果$a$和$b$是$S$的元素且$a < b$，则$b$不是$a$的倍数。$S$的一个元素的最小可能值为多少？

(A) 2 2

(B) 3 3

(D) 5 5

(E) 7 7

Answer

Correct choice: (C)

正确答案：（C）

Solution

If 1 $\in S$, then S can have only 1 element, not 6 elements. If 2 is the least element of S, then 2, 3, 5, 7, 9, and 11 are available to be in S, but 3 and 9 cannot both be in S, so the largest possible size of S is 5. If 3 is the least element, then 3, 4, 5, 7, 8, 10, and 11 are available, but at most one of 4 and 8 can be in S and at most one of 5 and 10 can be in S, so again S has size at most 5. The set S = ${4, 6, 7, 9, 10, 11}$ has the required property, so 4 is the least possible element of S.

如果1 $\in S$，则S只能有1个元素，不能有6个元素。如果2是S的最小元素，则可选择的数有2、3、5、7、9和11，但3和9不能同时在S中，因此S的最大可能大小是5。如果3是S的最小元素，则可选择的数有3、4、5、7、8、10和11，但4和8最多只能选一个，5和10最多只能选一个，因此S的大小至多为5。集合S = ${4, 6, 7, 9, 10, 11}$具有要求性质，因此4是S的最小可能元素。

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