If-x-y-Z-23-2-x-5-y-k-and-k-is-an-even-number-then-what-is-max-x-y-

Question Number 176847 by Ar Brandon last updated on 27/Sep/22

$$\mathrm{If}\:{x},\:{y}\:\in\mathbb{Z} \\ $$$$\mathrm{23}!=\mathrm{2}^{{x}} \mathrm{5}^{{y}} {k} \\ $$$$\mathrm{and}\:{k}\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{number},\:\mathrm{then}\:\mathrm{what}\:\mathrm{is}\: \\ $$$$\mathrm{max}\left({x}+{y}\right)\:? \\ $$

Answered by BaliramKumar last updated on 27/Sep/22

determinant ((2,(23)),(2,(11)),(2,5),(2,2),(2,1),(,0)) & determinant ((5,(23)),(5,4),(,0)) determinant ((),()) x = 11+5+2+1+0 = 19 y = 4+0 = 4 23! = 2^(19) ×5^4 ×x [where ′x′ will be odd] 23! = 2^(18) ×5^4 ×(2x) 23! = 2^(18) ×5^4 ×k [2x = k] max(x+y) = 18+4 = determinant (((22)))

$$ \\ $$$$\begin{array}{|c|c|c|c|c|c|}{\mathrm{2}}&\hline{\mathrm{23}}\\{\mathrm{2}}&\hline{\mathrm{11}}\\{\mathrm{2}}&\hline{\mathrm{5}}\\{\mathrm{2}}&\hline{\mathrm{2}}\\{\mathrm{2}}&\hline{\mathrm{1}}\\{}&\hline{\mathrm{0}}\\\hline\end{array}\:\:\:\:\:\:\&\:\:\:\:\:\:\begin{array}{|c|c|c|}{\mathrm{5}}&\hline{\mathrm{23}}\\{\mathrm{5}}&\hline{\mathrm{4}}\\{}&\hline{\mathrm{0}}\\\hline\end{array}\begin{array}{|c|c|}\\\\\hline\end{array} \\ $$$${x}\:=\:\mathrm{11}+\mathrm{5}+\mathrm{2}+\mathrm{1}+\mathrm{0}\:=\:\mathrm{19} \\ $$$${y}\:=\:\mathrm{4}+\mathrm{0}\:=\:\mathrm{4} \\ $$$$\mathrm{23}!\:=\:\mathrm{2}^{\mathrm{19}} ×\mathrm{5}^{\mathrm{4}} ×{x}\:\:\:\:\:\:\:\:\:\:\:\:\:\left[{where}\:'{x}'\:{will}\:{be}\:{odd}\right] \\ $$$$\mathrm{23}!\:=\:\mathrm{2}^{\mathrm{18}} ×\mathrm{5}^{\mathrm{4}} ×\left(\mathrm{2}{x}\right) \\ $$$$\mathrm{23}!\:=\:\mathrm{2}^{\mathrm{18}} ×\mathrm{5}^{\mathrm{4}} ×{k}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\mathrm{2}{x}\:=\:{k}\right] \\ $$$${max}\left({x}+{y}\right)\:=\:\mathrm{18}+\mathrm{4}\:=\:\begin{array}{|c|}{\mathrm{22}}\\\hline\end{array} \\ $$$$ \\ $$

Commented by Ar Brandon last updated on 28/Sep/22

Thanks

Answered by mr W last updated on 27/Sep/22

⌊((23)/2)⌋+⌊((23)/2^2 )⌋+⌊((23)/2^3 )⌋+⌊((23)/2^4 )⌋=11+5+2+1=19 ⌊((23)/5)⌋=4 ⇒23!=2^(19) ×5^4 ×... such that (...) is even, it must contain at least one factor 2. ⇒x_(max) =18, y_(max) =4 ⇒(x+y)_(max) =18+4=22

$$\lfloor\frac{\mathrm{23}}{\mathrm{2}}\rfloor+\lfloor\frac{\mathrm{23}}{\mathrm{2}^{\mathrm{2}} }\rfloor+\lfloor\frac{\mathrm{23}}{\mathrm{2}^{\mathrm{3}} }\rfloor+\lfloor\frac{\mathrm{23}}{\mathrm{2}^{\mathrm{4}} }\rfloor=\mathrm{11}+\mathrm{5}+\mathrm{2}+\mathrm{1}=\mathrm{19} \\ $$$$\lfloor\frac{\mathrm{23}}{\mathrm{5}}\rfloor=\mathrm{4} \\ $$$$\Rightarrow\mathrm{23}!=\mathrm{2}^{\mathrm{19}} ×\mathrm{5}^{\mathrm{4}} ×… \\ $$$${such}\:{that}\:\left(…\right)\:{is}\:{even},\:{it}\:{must}\:{contain} \\ $$$${at}\:{least}\:{one}\:{factor}\:\mathrm{2}. \\ $$$$\Rightarrow{x}_{{max}} =\mathrm{18},\:{y}_{{max}} =\mathrm{4} \\ $$$$\Rightarrow\left({x}+{y}\right)_{{max}} =\mathrm{18}+\mathrm{4}=\mathrm{22} \\ $$

Commented by Ar Brandon last updated on 27/Sep/22

Thank you ��

Answered by Rasheed.Sindhi last updated on 27/Sep/22

2 4 6 8 10 12 14 16 18 20 22......11 4 8 12 16 20..........5 8 16..................2 16.....................1 11+5+2+1=19 2′s k ieven maximum x is 18 5 10 15 20...........4 maximum y 4 max(x+y)=18+4=22

$$\mathrm{2}\:\mathrm{4}\:\mathrm{6}\:\mathrm{8}\:\mathrm{10}\:\mathrm{12}\:\mathrm{14}\:\mathrm{16}\:\mathrm{18}\:\mathrm{20}\:\mathrm{22}……\mathrm{11} \\ $$$$\mathrm{4}\:\mathrm{8}\:\mathrm{12}\:\mathrm{16}\:\mathrm{20}……….\mathrm{5} \\ $$$$\mathrm{8}\:\mathrm{16}………………\mathrm{2} \\ $$$$\mathrm{16}…………………\mathrm{1} \\ $$$$\mathrm{11}+\mathrm{5}+\mathrm{2}+\mathrm{1}=\mathrm{19}\:\mathrm{2}'{s} \\ $$$${k}\:{ieven}\:{maximum}\:{x}\:{is}\:\mathrm{18} \\ $$$$\mathrm{5}\:\mathrm{10}\:\mathrm{15}\:\mathrm{20}………..\mathrm{4}\: \\ $$$${maximum}\:{y}\:\:\mathrm{4} \\ $$$${max}\left({x}+{y}\right)=\mathrm{18}+\mathrm{4}=\mathrm{22} \\ $$

Commented by Ar Brandon last updated on 27/Sep/22

Thanks ��

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If-x-y-Z-23-2-x-5-y-k-and-k-is-an-even-number-then-what-is-max-x-y-

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