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How many integers n are there such that \(\dfrac{15 - n}{3 - n}\) is also an integer?

 May 24, 2020
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How many integers \(n\) are there such that \(\dfrac{15 - n}{3 - n}\) is also an integer?

\(\begin{array}{|rcll|} \hline \dfrac{15 - n}{3 - n} &=& \dfrac{12+3-n}{3 - n} \\\\ &=& \dfrac{12}{3 - n}+\dfrac{3-n}{3 - n} \\\\ \mathbf{\dfrac{15 - n}{3 - n}} &=& \mathbf{\dfrac{12}{3 - n}+1} \\ \hline \end{array} \)

 

Divisors of 12:
1 | 2 | 3 | 4 | 6 | 12   (6 divisors)

 

\(\begin{array}{|rcll|} \hline 3-n &=& 1 \qquad \text{or} \qquad \mathbf{n=2} \\ 3-n &=& 2 \qquad \text{or} \qquad \mathbf{n=1} \\ 3-n &=& 3 \qquad \text{or} \qquad \mathbf{n=0} \\ 3-n &=& 4 \qquad \text{or} \qquad \mathbf{n=-1} \\ 3-n &=& 6 \qquad \text{or} \qquad \mathbf{n=-3} \\ 3-n &=& 12 \qquad \text{or} \qquad \mathbf{n=-9} \\ \hline \end{array}\)

 

\(n=\{2,1,0,-1,-3,-9\}\)

 

laugh

 May 25, 2020

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