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Difference between revisions of "2014 AMC 10B Problems/Problem 12"

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==Problem==
 
==Problem==
  
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The largest divisor of <math>2,014,000,000</math> is itself. What is the fifth-largest divisor?
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The largest divisor of <math>2,014,000,000</math> is itself. What is its fifth-largest divisor?
  
 
<math> \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 </math>
 
<math> \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 </math>
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==Solution==
 
==Solution==
  
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Note that <math>2\ ,014\ ,000\ ,000</math> is divisible by <math>2,\ 4,\ 5,\ 8</math>. Then, the fifth largest factor would come from divisibility by 8, or <math>251\ ,750\ ,000</math>, or <math>\textbf{(C)}</math>.
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Note that <math>2,014,000,000</math> is divisible by <math>1,\ 2,\ 4,\ 5,\ 8</math>. Then, the fifth largest factor would come from divisibility by <math>8</math>, or <math>251,750,000</math>, or <math>\boxed{\textbf{(C)}}</math>.
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Alternative method to dividing: notice that <math>2,014,000,000</math> factorizes into <math>2 \cdot 19 \cdot 53</math> times <math>10^6</math>. Thus, the answer will have <math>7-3 = 4</math> powers of 2, which means there are <math>4</math> zeroes in the answer because each power of <math>2</math> adds a zero.
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2014|ab=B|num-b=11|num-a=13}}
 
{{AMC10 box|year=2014|ab=B|num-b=11|num-a=13}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 18:19, 5 October 2024

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.

See Also

2014 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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