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A band is marching in a rectangular formation with dimensions $n-2$ and $n + 8$. In the second stage of their performance, they re-arrange to form a different rectangle with dimensions $n$ and $2n - 3$, excluding all the drummers. If there are at least 4 drummers, then find the sum of all possible values of $n$.

 Mar 14, 2019

Best Answer 

 #1
avatar+6196 
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\(\text{There are }M=(n-2)(n+8) \text{ total marchers}\\ \text{let }D\text{ be the number of drummers},~D\geq 4\\ M-D = n(2n-3)\)

 

\(n^2+6n-16 = 2n^2 - 3n+D\\ n^2 -9n+(16+D)=0\\ n = \dfrac{9\pm \sqrt{81-4(16+D)}}{2} \in \mathbb{N}\)

 

\(81 - 4(16+D) = k^2,~k \in \mathbb{N}\\ \text{The only possible value for }D \text{ is 4}\\ n = \dfrac{9 \pm 1}{2} = 5,4\\ 4+5=9\)

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 Mar 14, 2019
 #1
avatar+6196 
+2
Best Answer

\(\text{There are }M=(n-2)(n+8) \text{ total marchers}\\ \text{let }D\text{ be the number of drummers},~D\geq 4\\ M-D = n(2n-3)\)

 

\(n^2+6n-16 = 2n^2 - 3n+D\\ n^2 -9n+(16+D)=0\\ n = \dfrac{9\pm \sqrt{81-4(16+D)}}{2} \in \mathbb{N}\)

 

\(81 - 4(16+D) = k^2,~k \in \mathbb{N}\\ \text{The only possible value for }D \text{ is 4}\\ n = \dfrac{9 \pm 1}{2} = 5,4\\ 4+5=9\)

Rom Mar 14, 2019

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