2021 AMC 10B Spring 第 12 题

Question

设

N = 34 \cdot 34 \cdot 63 \cdot 270

。

N

的奇因数之和与

N

的偶因数之和的比是多少？

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?

1 : 16

1 : 15

1 : 14

1 : 8

1 : 3

按因数中

2

的幂次把因数分开。

Separate divisors by their power of

2

每个奇因数都对应偶因数

2d,4d,8d

。

Each odd divisor corresponds to even divisors

2d,4d,8d

分解质因数得

N =2^3\cdot 3^5\cdot 5\cdot 7\cdot 17^2\text{。}

若

x

是

N

的一个奇因数，则

2x,4x,8x

都是

N

的偶因数，它们的和为

14x

。把所有奇因数都这样配对，偶因数之和就是奇因数之和的

14

倍。因此所求比为

1:14

。

所以答案是 C。

Using prime factorization, we get

N =2^3\cdot 3^5\cdot 5\cdot 7\cdot 17^2.

If we have an odd divisor

x

of

N,

then

2x,4x,8x

are divisors of

N,

which has a combined sum of

14x.

If we take the sum of every odd divisor, then the even divisors must have a sum which is

14

times the sum of the odd divisors. Therefore, the requested ratio is

1:14.

Thus, the answer is C .

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