Question

有多少个由三个正整数组成的集合 $\{a,b,c\}$ 满足 $a\cdot b\cdot c=2310$ ？

For how many three-element sets of positive integers $\{a,b,c\}$ is it true that $a\cdot b\cdot c=2310?$

$32$

$36$

$40$

$43$

$45$

答案：C

知识点：质因数分解 set partitions

难度评级：2280

小提示：

将 $2310$ 分解为互不相同的质因数，并把每个质因数分配给一个因数

Factor $2310$ into distinct primes and assign each prime to one of the factors

大提示：

分别统计一个因数为 $1$ 以及三个因数都大于 $1$ 的情形

Count separately the cases in which one factor is $1$ and in which all three factors exceed $1$

解答：

因为 $2310=2\cdot3\cdot5\cdot7\cdot11$ ，每个质因数恰好属于三个因数中的一个。若三个因数都大于 $1$ ，它们对应的无序质因数组就是把五个对象分成三个非空组，其数量为 $S(5,3)=\frac{3^5-3\cdot2^5+3}{6}=25\text{。}$ 若有一个因数为 $1$ ，则把这些质因数分成两个非空组，得到 $S(5,2)=2^4-1=15$ 。由于互不相交的质因数集合不同，所得因数也互不相同。因此总数为 $25+15=40$ ，正确答案是 C。

Since $2310=2\cdot3\cdot5\cdot7\cdot11,$ each prime belongs to exactly one of the three factors. If all factors exceed $1,$ their unordered prime groups form a partition of five objects into three nonempty blocks, counted by $S(5,3)=\frac{3^5-3\cdot2^5+3}{6}=25.$ If one factor is $1,$ the primes are partitioned into two nonempty blocks, giving $S(5,2)=2^4-1=15.$ The factors are distinct because their disjoint prime sets differ. Thus the total is $25+15=40,$ and the correct answer is C.

1995 AMC 12 第 29 题

答案：C

难度评级：2280

小提示：

大提示：

解答：

其他年份的第 29 题