find-all-solutions-of-the-equation-x-2-x-in-each-of-the-rings-Z-2-Z-3-and-Z-6-

Question Number 214001 by RoseAli last updated on 24/Nov/24

find all solutions of the equation x^{2} = x in each of the rings

Z_{2}

Z_{3}

and Z_{6}

Answered by TonyCWX08 last updated on 24/Nov/24

I n Z_{2}, Z e r o D i v i s o r : 1 \times 2 = 0

x^{2} - x = 0

x (x - 1) = 0

x = 0 o r x = 1

B u t 2 (2 - 1) = 2 \times 1 = 0

S o x = 2 i s a l s o a n a n s w e r

Answered by mehdee7396 last updated on 24/Nov/24

Z_{2} \Rightarrow 2 ∣ x (x - 1) \Rightarrow x \in Z

Z_{3} \Rightarrow 3 ∣ x (x - 1) \Rightarrow x = 3 k o r x = 3 k + 1

Z_{6} \Rightarrow 6 ∣ x (x - 1) \Rightarrow x = 3 k o r x = 3 k + 1