Question Number 106422 by Algoritm last updated on 05/Aug/20
Commented by mr W last updated on 05/Aug/20
$$\mathrm{73}!=\mathrm{1}×\mathrm{2}×\mathrm{3}×...×\mathrm{49}×\mathrm{50}×...×\mathrm{73} \\ $$$$\mathrm{73}!=\mathrm{0}\:\left({mod}\:\mathrm{2}\right) \\ $$$$\mathrm{73}!=\mathrm{0}\:\left({mod}\:\mathrm{3}\right) \\ $$$$\mathrm{73}!=\mathrm{0}\:\left({mod}\:\mathrm{4}\right) \\ $$$$... \\ $$$$\mathrm{73}!=\mathrm{0}\:\left({mod}\:\mathrm{49}\right) \\ $$$$\mathrm{73}!=\mathrm{0}\:\left({mod}\:\mathrm{50}\right) \\ $$$$... \\ $$$$\mathrm{73}!=\mathrm{0}\:\left({mod}\:\mathrm{73}\right) \\ $$
Answered by Olaf last updated on 24/Sep/20
![73! = 48!×49×((73!)/(49!)) 73! = 49k with k = ((73!)/(49)) ⇒73! ≡ 0 [49]](Q115185.png)
$$\mathrm{73}!\:=\:\mathrm{48}!×\mathrm{49}×\frac{\mathrm{73}!}{\mathrm{49}!} \\ $$$$\mathrm{73}!\:=\:\mathrm{49}{k}\:\mathrm{with}\:{k}\:=\:\frac{\mathrm{73}!}{\mathrm{49}} \\ $$$$\Rightarrow\mathrm{73}!\:\equiv\:\mathrm{0}\:\left[\mathrm{49}\right] \\ $$