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Question-146224




Question Number 146224 by mathdanisur last updated on 12/Jul/21
Answered by gsk2684 last updated on 12/Jul/21
1! 2! 3! ... (x−1)! =288=2^5 3^2   x−1≤4  observing 1! 2! 3! 4!=288  x=5
$$\mathrm{1}!\:\mathrm{2}!\:\mathrm{3}!\:…\:\left({x}−\mathrm{1}\right)!\:=\mathrm{288}=\mathrm{2}^{\mathrm{5}} \mathrm{3}^{\mathrm{2}} \\ $$$${x}−\mathrm{1}\leqslant\mathrm{4} \\ $$$${observing}\:\mathrm{1}!\:\mathrm{2}!\:\mathrm{3}!\:\mathrm{4}!=\mathrm{288} \\ $$$${x}=\mathrm{5} \\ $$
Commented by mathdanisur last updated on 12/Jul/21
cool Ser thanks
$${cool}\:{Ser}\:{thanks} \\ $$
Answered by Rasheed.Sindhi last updated on 12/Jul/21
 0!.1!.2!.3!....(x−1)!=288  ⇒ (x−1)! ∣ 288 ∧ x! ∤ 288⇒x−1=4  ⇒x=5
$$\:\mathrm{0}!.\mathrm{1}!.\mathrm{2}!.\mathrm{3}!….\left({x}−\mathrm{1}\right)!=\mathrm{288} \\ $$$$\Rightarrow\:\left({x}−\mathrm{1}\right)!\:\mid\:\mathrm{288}\:\wedge\:{x}!\:\nmid\:\mathrm{288}\Rightarrow{x}−\mathrm{1}=\mathrm{4} \\ $$$$\Rightarrow{x}=\mathrm{5} \\ $$
Commented by mathdanisur last updated on 12/Jul/21
Cool thanks Ser
$${Cool}\:{thanks}\:{Ser} \\ $$

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