2014 AMC 10B 第 12 题

Question

数

2{,}014{,}000{,}000

的最大因数是它本身。它的第五大因数是多少？

The largest divisor of

2{,}014{,}000{,}000

is itself. What is its fifth-largest divisor?

125{,}875{,}000

201{,}400{,}000

251{,}750{,}000

402{,}800{,}000

503{,}500{,}000

第五大因数与第五小因数配对。

The fifth largest divisor pairs with the fifth smallest divisor

从素因数分解中列出最小的几个因数。

List the smallest divisors from the prime factorization

第五大因数等于

2,014,000,000

除以第五小因数。

2,014,000,000

的素因数分解为

2^7\cdot 5^6 \cdot 19\cdot 53\text{。}

最小的

5

个因数是

1,2,4,5,8

。所以第五小因数为

8

，第五大因数为

\dfrac{2014000000}8 =251750000\text{。}

所以正确答案是 C。

The fifth-largest divisor is

2,014,000,000

divided by the fifth smallest divisor.

The prime factorization of

2,014,000,000

is:

2^7\cdot 5^6 \cdot 19\cdot 53.

This makes the first

5

smallest divisors

1,2,4,5,8.

Therefore, the fifth smallest divisor is

8,

and the fifth largest divisor must be:

\dfrac{2014000000}8 =251750000.

Thus, the correct answer is C .

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