AMC10 2016 A

AMC10 2016 A · Q7

AMC10 2016 A · Q7. It mainly tests Averages (mean), Casework.

For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?

对于某个正整数 $n$，数 $110n^3$ 有 $110$ 个正整数因数（包括 $1$ 和 $110n^3$ 本身）。那么数 $81n^4$ 有多少个正整数因数？

(A) 50 50

(B) 60 60

(D) 90 90

(E) 100 100

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): The mean of the data values is \[ \frac{60+100+x+40+50+200+90}{7}=\frac{x+540}{7}=x. \] Solving this equation for $x$ gives $x=90$. Thus the data in nondecreasing order are $40,50,60,90,90,100,200$, so the median is $90$ and the mode is $90$, as required.

答案（D）：数据的平均值为 \[ \frac{60+100+x+40+50+200+90}{7}=\frac{x+540}{7}=x. \] 解此方程得 $x=90$。因此，按非递减顺序排列的数据为 $40,50,60,90,90,100,200$，所以中位数是 $90$，众数是 $90$，符合要求。

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