Is-this-proposition-true-x-Z-x-2-x-3-0-5-if-and-only-if-x-1-5-

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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