Is this proposition true?: ∀ x ∈ Z, x^2 +x+3≡0[5] if and only if x≡1[5]


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Question Number 134036 by mathocean1 last updated on 26/Feb/21

$I s t h i s p r o p o s i t i o n t r u e ? :$ $\forall x \in Z, x^{2} + x + 3 \equiv 0 [5] i f a n d o n l y i f$ $x \equiv 1 [5]$
Commented by mr W last updated on 27/Feb/21

${i t}^{'} s n o t t r u e!$ $e x a m p l e s :$ $x = 3 ≢ 1 [5]$ $b u t x^{2} + x + 3 = 15 \equiv 0 [5]$ $x = 8 ≢ 1 [5]$ $b u t x^{2} + x + 3 = 75 \equiv 0 [5]$ $g e n e r a l l y w i t h$ $x = 5 k + 3 ≢ 1 [5]$ $b u t x^{2} + x + 3 = 5 (5 k^{2} + 7 k + 3) \equiv 0 [5]$
Commented by mr W last updated on 27/Feb/21

$\forall x \in Z, x^{2} + x + 3 \equiv 0 [5] i f a n d o n l y i f$ $x \equiv 1 [5]$
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