Menu Close

x-y-N-0-1-x-y-find-z-N-with-z-x-y-




Question Number 94786 by MJS last updated on 21/May/20
x, y ∈N\{0, 1} ∧ x≤y  find z∈N with z!=x!y!
$${x},\:{y}\:\in\mathbb{N}\backslash\left\{\mathrm{0},\:\mathrm{1}\right\}\:\wedge\:{x}\leqslant{y} \\ $$$$\mathrm{find}\:{z}\in\mathbb{N}\:\mathrm{with}\:{z}!={x}!{y}! \\ $$
Commented by hknkrc46 last updated on 21/May/20
(a) x!=y+1 ⇒ z!=(y+1)! ⇒ z=y+1   (a.1) x=3⇒y=5⇒z=6  (a.2) x=4⇒y=23⇒z=24  (a.3) x=5⇒y=119⇒z=120  ∙∙∙∙  {x,y,z : x!=y+1=z ; x≥3}  (b) y!=x+1 ⇒ z!=(x+1)! ⇒ z=x+1  (b.1) y=3⇒x=5⇒z=6 (x≰y)  (b.2) y=4⇒x=23⇒z=24 (x≰y)  ....
$$\left(\mathrm{a}\right)\:\mathrm{x}!=\mathrm{y}+\mathrm{1}\:\Rightarrow\:\mathrm{z}!=\left(\mathrm{y}+\mathrm{1}\right)!\:\Rightarrow\:\mathrm{z}=\mathrm{y}+\mathrm{1}\: \\ $$$$\left(\mathrm{a}.\mathrm{1}\right)\:\mathrm{x}=\mathrm{3}\Rightarrow\mathrm{y}=\mathrm{5}\Rightarrow\mathrm{z}=\mathrm{6} \\ $$$$\left(\mathrm{a}.\mathrm{2}\right)\:\mathrm{x}=\mathrm{4}\Rightarrow\mathrm{y}=\mathrm{23}\Rightarrow\mathrm{z}=\mathrm{24} \\ $$$$\left(\mathrm{a}.\mathrm{3}\right)\:\mathrm{x}=\mathrm{5}\Rightarrow\mathrm{y}=\mathrm{119}\Rightarrow\mathrm{z}=\mathrm{120} \\ $$$$\centerdot\centerdot\centerdot\centerdot \\ $$$$\left\{\mathrm{x},\mathrm{y},\mathrm{z}\::\:\mathrm{x}!=\mathrm{y}+\mathrm{1}=\mathrm{z}\:;\:\mathrm{x}\geqslant\mathrm{3}\right\} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{y}!=\mathrm{x}+\mathrm{1}\:\Rightarrow\:\mathrm{z}!=\left(\mathrm{x}+\mathrm{1}\right)!\:\Rightarrow\:\mathrm{z}=\mathrm{x}+\mathrm{1} \\ $$$$\left(\mathrm{b}.\mathrm{1}\right)\:\mathrm{y}=\mathrm{3}\Rightarrow\mathrm{x}=\mathrm{5}\Rightarrow\mathrm{z}=\mathrm{6}\:\left(\mathrm{x}\nleq\mathrm{y}\right) \\ $$$$\left(\mathrm{b}.\mathrm{2}\right)\:\mathrm{y}=\mathrm{4}\Rightarrow\mathrm{x}=\mathrm{23}\Rightarrow\mathrm{z}=\mathrm{24}\:\left(\mathrm{x}\nleq\mathrm{y}\right) \\ $$$$…. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *