AMC12 2021 B

AMC12 2021 B · Q7

AMC12 2021 B · Q7. It mainly tests Primes & prime factorization, Counting divisors.

Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$?

设$N = 34 \cdot 34 \cdot 63 \cdot 270$。$N$的奇约数之和与偶约数之和的比值为多少？

(A) 1 : 16 1 : 16

(B) 1 : 15 1 : 15

(D) 1 : 8 1 : 8

(E) 1 : 3 1 : 3

Answer

Correct choice: (C)

正确答案：（C）

Solution

Prime factorize $N$ to get $N=2^{3} \cdot 3^{5} \cdot 5\cdot 7\cdot 17^{2}$. For each odd divisor $n$ of $N$, there exist even divisors $2n, 4n, 8n$ of $N$, therefore the ratio is $1:(2+4+8)=\boxed{\textbf{(C)} ~1 : 14}$

将$N$质因数分解得到$N=2^{3} \cdot 3^{5} \cdot 5\cdot 7\cdot 17^{2}$。对于$N$的每个奇约数$n$，存在偶约数$2n, 4n, 8n$，因此比值为$1:(2+4+8)=\boxed{\textbf{(C)} ~1 : 14}$

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