How many factors of 2020 have more than 3 factors? (As an example, 12 has 6 factors, namely 1, 2, 3, 4, 6, and 12.)

A. 6

B. 7

C. 8

D. 9

E. 10

1. 2020 AMC 8 官方解析

The number 2020 has 12 factors.
$2020 = 1 \cdot 2020$
$= 2 \cdot 1010$
$= 4 \cdot 505$
$= 5 \cdot 404$
$= 10 \cdot 202$
$= 20 \cdot 101$

These factors may be classified as follows:
• The number 1 has exactly 1 factor.
• The numbers 2, 5, and 101 are primes, so each has exactly 2 factors.
• The number 4 has exactly 3 factors, namely 1, 2, and 4.
• The remaining 7 numbers, 10, 20, 202, 404, 505, 1010, and 2020, each have more than 3 factors. Thus 7 of the factors of 2020 have more than 3 factors.

2020 有 12 个约数 .
$2020 = 1 \cdot 2020$
$= 2 \cdot 1010$
$= 4 \cdot 505$
$= 5 \cdot 404$
$= 10 \cdot 202$
$= 20 \cdot 101$

这些约数可以按照如下分类：
• 数字1有且只有1个约数 .
• 数字2 , 5和101都是质数，所有每个数都有且只有2个约数 .
• 数字4有且只有3个约数，分别是1 , 2和4 .
• 其余7个数字，10 , 20 , 202 , 404 , 505 , 1010 , 和2020，每个数都有3个以上的约数 .
因此2020的约数中有7个数具有3个以上的约数 .

In order to determine the number of factors of 2020, note that the prime factorization of 2020 is $2020 = 2^2 \cdot 5^1 \cdot 101^1$.
This means that each factor of 2020 has the form $2^a \cdot 5^b \cdot 101^c$, where $a = 0, 1$, or $2$; $b = 0$ or $1$; and $c = 0$ or $1$. Thus the number of factors of 2020 is $3 \cdot 2 \cdot 2 = 12$.
By similar reasoning, any positive integer with at least 2 distinct prime factors, say $p$ and $q$, has at least 4 factors, namely $1$, $p$, $q$, and $pq$.
The factors of 2020 that have no more than 1 distinct prime factor are 1, 2, 4, 5, and 101. Each of these has at most 3 factors. The remaining 7 factors, 10, 20, 202, 404, 505, 1010, and 2020, each have at least 2 distinct prime factors. Therefore 7 of the factors of 2020 have more than 3 factors.

为了确定2020的所有约数的数量，可以将2020进行素因数分解如下
$2020 = 2^2 \cdot 5^1 \cdot 101^1$
这意味着2020的每个约数都可以表示为$2^a \cdot 5^b \cdot 101^c$的形式，其中$a = 0, 1$或$2$；$b = 0$或$1$；并且$c = 0$或$1$. 因此，2020的约数数量为$3 \cdot 2 \cdot 2 = 12$.
类似地，任何具有至少2个不同的素数因子（例如$p$和$q$）的正整数都具有至少4个约数，即$1$，$p$，$q$和$pq$. 2020的约数中具有不超过1个素数因子的是$1, 2, 4, 5$和$101$. 其中每个数最多具有3个约数 . 其余的7个因子，即$10, 20, 202, 404, 505, 1010$和$2020$，每个数至少具有2个素数因子 . 因此，在2020的所有约数中有7个数有3个以上的约数 .