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Question Number 71926 by TawaTawa last updated on 22/Oct/19

Commented by Prithwish sen last updated on 22/Oct/19

If !! means mere factorial than 3002!! must be divisible by 2003 i.e 3092!! ≡ 0 (mod 2003)

$$\boldsymbol{\mathrm{If}}\:\:!!\:\:\boldsymbol{\mathrm{means}}\:\boldsymbol{\mathrm{mere}}\:\boldsymbol{\mathrm{factorial}}\:\boldsymbol{\mathrm{than}}\:\mathrm{3002}!!\:\boldsymbol{\mathrm{must}} \\ $$$$\boldsymbol{\mathrm{be}}\:\boldsymbol{\mathrm{divisible}}\:\boldsymbol{\mathrm{by}}\:\mathrm{2003} \\ $$$$\boldsymbol{\mathrm{i}}.\boldsymbol{\mathrm{e}} \\ $$$$\:\:\:\:\:\mathrm{3092}!!\:\equiv\:\mathrm{0}\:\left(\boldsymbol{\mathrm{mod}}\:\mathrm{2003}\right) \\ $$

Commented by TawaTawa last updated on 23/Oct/19

God bless you sirs

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sirs} \\ $$

Answered by MJS last updated on 22/Oct/19

{ ((n=2k ⇒ n!!=2^k k!)),((n=2k−1 ⇒ n!!=(((2k)!)/(2^k k!)))) :} definition: (−1)!!=0!!=1 ⇒ 3002!!=2^(1501) 1501!

$$\begin{cases}{{n}=\mathrm{2}{k}\:\Rightarrow\:{n}!!=\mathrm{2}^{{k}} {k}!}\\{{n}=\mathrm{2}{k}−\mathrm{1}\:\Rightarrow\:{n}!!=\frac{\left(\mathrm{2}{k}\right)!}{\mathrm{2}^{{k}} {k}!}}\end{cases} \\ $$$$\mathrm{definition}:\:\left(−\mathrm{1}\right)!!=\mathrm{0}!!=\mathrm{1} \\ $$$$\Rightarrow \\ $$$$\mathrm{3002}!!=\mathrm{2}^{\mathrm{1501}} \mathrm{1501}! \\ $$

Commented by MJS last updated on 22/Oct/19

sorry it′s not an answer, just an explanation

$$\mathrm{sorry}\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{an}\:\mathrm{answer},\:\mathrm{just}\:\mathrm{an}\:\mathrm{explanation} \\ $$

Commented by Prithwish sen last updated on 22/Oct/19

Thank you sir . But what it is called sir ?

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}\:.\:\mathrm{But}\:\mathrm{what}\:\mathrm{it}\:\mathrm{is}\:\mathrm{called}\:\mathrm{sir}\:? \\ $$

Commented by MJS last updated on 22/Oct/19

“double faculty” n!!=n(n−2)(n−4)...

$$``\mathrm{double}\:\mathrm{faculty}'' \\ $$$${n}!!={n}\left({n}−\mathrm{2}\right)\left({n}−\mathrm{4}\right)... \\ $$

Commented by Prithwish sen last updated on 22/Oct/19

that means 102!! = 2.4.6.......102 101!! = 1.3.5......101 102! = 101!! . 102!! Wow! Thank you very much sir. I learn something new .

$$\mathrm{that}\:\mathrm{means} \\ $$$$\mathrm{102}!!\:=\:\mathrm{2}.\mathrm{4}.\mathrm{6}.......\mathrm{102} \\ $$$$\mathrm{101}!!\:=\:\mathrm{1}.\mathrm{3}.\mathrm{5}......\mathrm{101} \\ $$$$\mathrm{102}!\:=\:\mathrm{101}!!\:.\:\mathrm{102}!! \\ $$$$\mathrm{Wow}!\:\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{learn}\:\mathrm{something} \\ $$$$\mathrm{new}\:. \\ $$

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