Question Number 134036 by mathocean1 last updated on 26/Feb/21
![Is this proposition true?: ∀ x ∈ Z, x^2 +x+3≡0[5] if and only if x≡1[5]](https://www.tinkutara.com/question/Q134036.png)
$${Is}\:{this}\:{proposition}\:{true}?: \\ $$$$ \\ $$$$\forall\:{x}\:\in\:\mathbb{Z},\:{x}^{\mathrm{2}} +{x}+\mathrm{3}\equiv\mathrm{0}\left[\mathrm{5}\right]\:{if}\:\:{and}\:{only}\:{if} \\ $$$$\:{x}\equiv\mathrm{1}\left[\mathrm{5}\right] \\ $$
Commented by mr W last updated on 27/Feb/21
![it′s not true! examples: x=3≢1 [5] but x^2 +x+3=15≡0 [5] x=8≢1 [5] but x^2 +x+3=75≡0 [5] generally with x=5k+3≢1 [5] but x^2 +x+3=5(5k^2 +7k+3)≡0 [5]](https://www.tinkutara.com/question/Q134054.png)
$${it}'{s}\:{not}\:{true}! \\ $$$${examples}: \\ $$$${x}=\mathrm{3}≢\mathrm{1}\:\left[\mathrm{5}\right] \\ $$$${but}\:{x}^{\mathrm{2}} +{x}+\mathrm{3}=\mathrm{15}\equiv\mathrm{0}\:\left[\mathrm{5}\right] \\ $$$${x}=\mathrm{8}≢\mathrm{1}\:\left[\mathrm{5}\right] \\ $$$${but}\:{x}^{\mathrm{2}} +{x}+\mathrm{3}=\mathrm{75}\equiv\mathrm{0}\:\left[\mathrm{5}\right] \\ $$$${generally}\:{with} \\ $$$${x}=\mathrm{5}{k}+\mathrm{3}≢\mathrm{1}\:\left[\mathrm{5}\right] \\ $$$${but}\:{x}^{\mathrm{2}} +{x}+\mathrm{3}=\mathrm{5}\left(\mathrm{5}{k}^{\mathrm{2}} +\mathrm{7}{k}+\mathrm{3}\right)\equiv\mathrm{0}\:\left[\mathrm{5}\right] \\ $$
Commented by mr W last updated on 27/Feb/21
![∀ x ∈ Z, x^2 +x+3≡0[5] if and only if x≡1[5]](https://www.tinkutara.com/question/Q134056.png)
$$\forall\:{x}\:\in\:\mathbb{Z},\:{x}^{\mathrm{2}} +{x}+\mathrm{3}\equiv\mathrm{0}\left[\mathrm{5}\right]\:{if}\:\cancel{\:{and}\:{only}\:{if}} \\ $$$$\:{x}\equiv\mathrm{1}\left[\mathrm{5}\right] \\ $$