Difference between revisions of "2017 AMC 10A Problems/Problem 16"

Line 2: Line 2:
 
There are <math>10</math> horses, named Horse <math>1</math>, Horse <math>2</math>, . . . , Horse <math>10</math>. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse <math>k</math> runs one lap in exactly <math>k</math> minutes. At time <math>0</math> 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 <math>S > 0</math>, in minutes, at which all <math>10</math> horses will again simultaneously be at the starting point is <math>S=2520</math>. Let <math>T > 0</math> 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 <math>T?</math>
 
There are <math>10</math> horses, named Horse <math>1</math>, Horse <math>2</math>, . . . , Horse <math>10</math>. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse <math>k</math> runs one lap in exactly <math>k</math> minutes. At time <math>0</math> 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 <math>S > 0</math>, in minutes, at which all <math>10</math> horses will again simultaneously be at the starting point is <math>S=2520</math>. Let <math>T > 0</math> 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 <math>T?</math>
  
<math>\textbf{(A) }2 \textbf{(B) }3 \textbf{(C) }4\textbf{(D) }5\textbf{(E) }6</math>
+
<math>\textbf{(A) }2 \qquad \textbf{(B) }3 \qquad \textbf{(C) }4 \qquad \textbf{(D) }5 \qquad \textbf{(E) }6</math>
  
 
==Solution==
 
==Solution==
  
 
==See Also==
 
==See Also==

Revision as of 18:28, 8 February 2017

Problem

There are $10$ horses, named Horse $1$, Horse $2$, . . . , Horse $10$. They get their names from how 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 again 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

See Also

Invalid username
Login to AoPS