AMC10 2008 A

AMC10 2008 A · Q3

AMC10 2008 A · Q3. It mainly tests Counting divisors.

For the positive integer $n$, let $\langle n\rangle$ denote the sum of all the positive divisors of $n$ with the exception of $n$ itself. For example, $\langle 4\rangle = 1 + 2 = 3$ and $\langle 12\rangle = 1 + 2 + 3 + 4 + 6 = 16$. What is $\langle\langle\langle 6\rangle\rangle\rangle$?

对于正整数$n$，令$\langle n\rangle$表示$n$本身以外的所有正因数之和。例如，$\langle 4\rangle = 1 + 2 = 3$，$\langle 12\rangle = 1 + 2 + 3 + 4 + 6 = 16$。求$\langle\langle\langle 6\rangle\rangle\rangle$。

(A) 6 6

(B) 12 12

(D) 32 32

(E) 36 36

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): The positive divisors of 6, other than 6, are 1, 2, and 3, so $<6> = 1 + 2 + 3 = 6$. As a consequence, we also have $<<6>> = 6$. Note: A positive integer whose divisors other than itself add up to that positive integer is called a perfect number. The two smallest perfect numbers are 6 and 28.

答案（A）：6 的正因数（不包括 6 本身）是 1、2 和 3，因此 $<6> = 1 + 2 + 3 = 6$。由此可得，我们也有 $<<6>> = 6$。注：若一个正整数的因数（除其自身外）之和等于该正整数，则称其为完全数。最小的两个完全数是 6 和 28。

Topics

NT.011 Counting divisors