Difference between revisions of "2014 AMC 10B Problems/Problem 12"
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==Problem== | ==Problem== | ||
| − | The largest divisor of <math>2,014,000,000</math> is itself. What is | + | 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== | ||
| − | 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>. | + | 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 
is itself. What is its fifth-largest divisor?
Solution
Note that is divisible by 
. Then, the fifth largest factor would come from divisibility by 
, or 
, or 
.
Alternative method to dividing: notice that factorizes into 
times 
. Thus, the answer will have 
powers of 2, which means there are 
zeroes in the answer because each power of 
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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