AMC8 2018

AMC8 2018 · Q18

AMC8 2018 · Q18. It mainly tests Primes & prime factorization, Counting divisors.

How many positive factors does 23,232 have?

23,232 有多少个正因数？

(A) 9 9

(B) 12 12

(D) 36 36

(E) 42 42

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): The prime factorization of $23{,}232$ is $2^6\cdot3\cdot11^2$. Each factor of $23{,}232$ must be of the form $2^a\cdot3^b\cdot11^c$, where $a=0,1,2,3,4,5,$ or $6$, $b=0$ or $1$, and $c=0,1,$ or $2$. Therefore, the number of factors of $23{,}232$ is $7\cdot2\cdot3=42$.

答案（E）：$23{,}232$ 的质因数分解为 $2^6\cdot3\cdot11^2$。$23{,}232$ 的每个因数都必须形如 $2^a\cdot3^b\cdot11^c$，其中 $a=0,1,2,3,4,5$ 或 $6$，$b=0$ 或 $1$，$c=0,1$ 或 $2$。因此，$23{,}232$ 的因数个数为 $7\cdot2\cdot3=42$。

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