2014 AMC 10B Problems/Problem 12

Problem

The largest divisor of $2,014,000,000$ is itself. What is its fifth-largest divisor?

$\textbf {(A) } 125, 875, 000 \qquad \textbf {(B) } 201, 400, 000 \qquad \textbf {(C) } 251, 750, 000 \qquad \textbf {(D) } 402, 800, 000 \qquad \textbf {(E) } 503, 500, 000$

Solution

Note that $2,014,000,000$ is divisible by $1,\ 2,\ 4,\ 5,\ 8$ . Then, the fifth largest factor would come from divisibility by $8$ , or $251,750,000$ , or $\boxed{\textbf{(C)}}$ .

Alternative method to dividing: notice that $2,014,000,000$ factorizes into $2 \cdot 19 \cdot 53$ times $10^6$ . Thus, the answer will have $7-3 = 4$ powers of 2, which means there are $4$ zeroes in the answer because each power of $2$ adds a zero.

2014 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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2014 AMC 10B Problems/Problem 12

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