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Compute $$\left\lfloor \frac{2007! + 2004!}{2006! + 2005!}\right\rfloor.$$

Aug 8, 2019

#1
+6203
+1

$$\left \lfloor \dfrac{2007! + 2004!}{2006!+2005!}\right\rfloor = \\ \left \lfloor \dfrac{2004!(1+2005\cdot 2006 \cdot 2007)}{2004!\cdot 2005(1+2006)}\right\rfloor = \\ \left \lfloor \dfrac{1+2005\cdot 2006 \cdot 2007}{2005(1+2006)}\right\rfloor = \\ \left \lfloor \dfrac{\frac{1}{2005}+2006 \cdot 2007}{1+2006}\right\rfloor = \\ \left \lfloor \dfrac{\frac{1}{2005\cdot 2006}+2007}{\frac{1}{2006}+1}\right\rfloor = 2006\\$$

The last equality comes from the first term in the numerator being so small we can ignore it,

and the denominator being slightly greater than 1 so that the floor is 2006 rather than 2007.

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Aug 9, 2019
edited by Rom  Aug 9, 2019
#2
+25648
+2

Compute  $$\Bigg\lfloor \dfrac{2007! + 2004!}{2006! + 2005!}\Bigg\rfloor$$

$$\begin{array}{|rcll|} \hline &&\mathbf{ \Bigg\lfloor \dfrac{2007! + 2004!}{2006! + 2005!}\Bigg \rfloor }\\ &=& \Bigg\lfloor \dfrac{2007!}{2006! + 2005!} + \dfrac{2004!}{2006! + 2005!} \Bigg \rfloor \\ &=& \Bigg\lfloor \dfrac{2006!~2007}{2005!~(1+2006)} + \dfrac{2004!}{2006! + 2005!} \Bigg \rfloor \\ &=& \Bigg\lfloor \dfrac{2006!~2007}{2005!~2007} + \dfrac{2004!}{2006! + 2005!} \Bigg \rfloor \\ &=& \Bigg\lfloor \dfrac{2006! }{2005! } + \dfrac{2004!}{2006! + 2005!} \Bigg \rfloor \\ &=& \Bigg\lfloor \dfrac{2005!~2006 }{2005! } + \dfrac{2004!}{2006! + 2005!} \Bigg \rfloor \\ &=& \Bigg\lfloor 2006 + \underbrace{\dfrac{2004!}{2006! + 2005!}}_{<1} \Bigg \rfloor \\ &=& \mathbf{2006} \\ \hline \end{array}$$

Aug 9, 2019