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Difference between revisions of "2017 AMC 12A Problems/Problem 12"

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==Solution==
 
==Solution==
  
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We know that Horse <math>k</math> will be at the starting point after <math>n</math> minutes iff <math>k|n</math>. Thus, we are looking for the smallest <math>n</math> such that at least <math>5</math> of the numbers <math>\{1,2,\cdots,10\}</math> divide <math>n</math>. Thus, <math>n</math> has at least <math>5</math> positive integer divisors.
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We know that Horse <math>k</math> will be at the starting point after <math>n</math> minutes if <math>k|n</math>. Thus, we are looking for the smallest <math>n</math> such that at least <math>5</math> of the numbers <math>\{1,2,\cdots,10\}</math> divide <math>n</math>. Thus, <math>n</math> has at least <math>5</math> positive integer divisors.
  
 
We quickly see that <math>12</math> is the smallest number with at least <math>5</math> positive integer divisors, and that <math>1,2,3,4,6</math> are each numbers of horses. Thus, our answer is <math>1+2=\boxed{\textbf{(B) } 3}</math>.
 
We quickly see that <math>12</math> is the smallest number with at least <math>5</math> positive integer divisors, and that <math>1,2,3,4,6</math> are each numbers of horses. Thus, our answer is <math>1+2=\boxed{\textbf{(B) } 3}</math>.

Revision as of 19:16, 8 February 2017

Problem

There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from ow many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will gain simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Solution

We know that Horse $k$ will be at the starting point after $n$ minutes if $k|n$. Thus, we are looking for the smallest $n$ such that at least $5$ of the numbers $\{1,2,\cdots,10\}$ divide $n$. Thus, $n$ has at least $5$ positive integer divisors.

We quickly see that $12$ is the smallest number with at least $5$ positive integer divisors, and that $1,2,3,4,6$ are each numbers of horses. Thus, our answer is $1+2=\boxed{\textbf{(B) } 3}$.

See Also

2017 AMC 12A (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 12 Problems and Solutions

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