AMC8 2020

AMC8 2020 · Q17

AMC8 2020 · Q17. It mainly tests Primes & prime factorization, Counting divisors.

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

$2020$ 有多少个正整数因数具有超过 $3$ 个因数？（例如，$12$ 有 $6$ 个因数，即 $1,2,3,4,6,$ 和 $12$。）

(A) 6 6

(B) 7 7

(D) 9 9

(E) 10 10

Answer

Correct choice: (B)

正确答案：（B）

Solution

Since $2020 = 2^2 \cdot 5 \cdot 101$, we can simply list its factors: \[1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020.\] There are $12$ factors; only $1, 2, 4, 5, 101$ don't have over $3$ factors, so the remaining $12-5 = \boxed{\textbf{(B) }7}$ factors have more than $3$ factors.

由于 $2020 = 2^2 \cdot 5 \cdot 101$，我们可以简单列出其因数：\[1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010, 2020.\] 有 $12$ 个因数；只有 $1, 2, 4, 5, 101$ 不超过 $3$ 个因数，所以剩余 $12-5 = \boxed{\textbf{(B) }7}$ 个因数具有超过 $3$ 个因数。

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