Question Number 194295 by Abdullahrussell last updated on 02/Jul/23
Answered by BaliramKumar last updated on 02/Jul/23
![1!×2!×3!×..................×2023! 1^(2023) ×2^(2022) ×3^(2021) ×..................×2023^1 1^(2023) ×...×5^(2019) ×....×10^(2014) ×..................×2020^4 ×...×2023^1 5^x ×y answer = x 5^1 ⇒ 2019+2014+.......+4 = (n/2)[a+a_n ] = ((404)/2)[4+2019] 202×2023 = 408646 5^2 ⇒ 25^(1999) +50^(1974) +..........+2000^(24) 1999+1974+......+24 = 80920 5^3 ⇒ 125^(1899) +250^(1774) +.....+2000^(24) 1899+1774+....+24 = 15384 5^4 ⇒ 625^(1399) +1250^(774) +1875^(149) 1399+774+149 = 2322 x = 408646+80920+15384+2322 determinant (((x = 507272)))](Q194298.png)
$$\mathrm{1}!×\mathrm{2}!×\mathrm{3}!×..................×\mathrm{2023}! \\ $$$$\mathrm{1}^{\mathrm{2023}} ×\mathrm{2}^{\mathrm{2022}} ×\mathrm{3}^{\mathrm{2021}} ×..................×\mathrm{2023}^{\mathrm{1}} \\ $$$$\mathrm{1}^{\mathrm{2023}} ×...×\mathrm{5}^{\mathrm{2019}} ×....×\mathrm{10}^{\mathrm{2014}} ×..................×\mathrm{2020}^{\mathrm{4}} ×...×\mathrm{2023}^{\mathrm{1}} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{5}^{{x}} ×{y} \\ $$$${answer}\:=\:{x} \\ $$$$\mathrm{5}^{\mathrm{1}} \Rightarrow \\ $$$$\mathrm{2019}+\mathrm{2014}+.......+\mathrm{4}\:=\:\frac{{n}}{\mathrm{2}}\left[{a}+{a}_{{n}} \right]\:=\:\frac{\mathrm{404}}{\mathrm{2}}\left[\mathrm{4}+\mathrm{2019}\right]\:\:\:\:\:\: \\ $$$$\mathrm{202}×\mathrm{2023}\:=\:\mathrm{408646} \\ $$$$\mathrm{5}^{\mathrm{2}} \Rightarrow \\ $$$$\mathrm{25}^{\mathrm{1999}} +\mathrm{50}^{\mathrm{1974}} +..........+\mathrm{2000}^{\mathrm{24}} \\ $$$$\mathrm{1999}+\mathrm{1974}+......+\mathrm{24}\:=\:\mathrm{80920} \\ $$$$\mathrm{5}^{\mathrm{3}} \Rightarrow \\ $$$$\mathrm{125}^{\mathrm{1899}} +\mathrm{250}^{\mathrm{1774}} +.....+\mathrm{2000}^{\mathrm{24}} \: \\ $$$$\mathrm{1899}+\mathrm{1774}+....+\mathrm{24}\:=\:\mathrm{15384} \\ $$$$\mathrm{5}^{\mathrm{4}} \Rightarrow \\ $$$$\mathrm{625}^{\mathrm{1399}} +\mathrm{1250}^{\mathrm{774}} +\mathrm{1875}^{\mathrm{149}} \\ $$$$\mathrm{1399}+\mathrm{774}+\mathrm{149}\:=\:\mathrm{2322} \\ $$$${x}\:=\:\mathrm{408646}+\mathrm{80920}+\mathrm{15384}+\mathrm{2322} \\ $$$$\begin{array}{|c|}{{x}\:=\:\mathrm{507272}}\\\hline\end{array} \\ $$