Difference between revisions of "2016 AMC 12B Problems/Problem 24"

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=Problem=
 
=Problem=
There are exactly <math>77,000</math> ordered quadruplets <math>(a, b, c, d)</math> such that <math>GCD(a, b, c, d) = 77</math> and <math>LMC(a, b, c, d) = n</math>. What is the smallest possible value for <math>n</math>?
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There are exactly <math>77,000</math> ordered quadruplets <math>(a, b, c, d)</math> such that <math>GCD(a, b, c, d) = 77</math> and <math>LCM(a, b, c, d) = n</math>. What is the smallest possible value for <math>n</math>?
  
 
<math>\textbf{(A)}\ 13,860\qquad\textbf{(B)}\ 20,790\qquad\textbf{(C)}\ 21,560 \qquad\textbf{(D)} 27,720 \qquad\textbf{(E)}\ 41,580</math>
 
<math>\textbf{(A)}\ 13,860\qquad\textbf{(B)}\ 20,790\qquad\textbf{(C)}\ 21,560 \qquad\textbf{(D)} 27,720 \qquad\textbf{(E)}\ 41,580</math>
  
 
=Solution=
 
=Solution=

Revision as of 14:46, 21 February 2016

Problem

There are exactly $77,000$ ordered quadruplets $(a, b, c, d)$ such that $GCD(a, b, c, d) = 77$ and $LCM(a, b, c, d) = n$. What is the smallest possible value for $n$?

$\textbf{(A)}\ 13,860\qquad\textbf{(B)}\ 20,790\qquad\textbf{(C)}\ 21,560 \qquad\textbf{(D)} 27,720 \qquad\textbf{(E)}\ 41,580$

Solution