2014 AMC 10B Problem 12

Question

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

The fifth largest divisor pairs with the fifth smallest divisor

List the smallest divisors from the prime factorization

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 .

Problem 12 in other years