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**Sampras** Suppose you have equations like $\displaystyle x+x+x+x+x = 0 $ in $\displaystyle \mathbb{Z}_5 $.

$\displaystyle \mathbb{Z}_5 = \{[0], [1], [2], [3], [4] \} $. So if we pick any representative from these congruence classes they will solve the equation. So $\displaystyle x = 0,1,2,3,4 $. But $\displaystyle x = 5,6,7,8,9 $ will also work right? Because they are representatives? But these values are "equal" to the other values.

Also the $\displaystyle x $ is just a symbol so could we just do the following: $\displaystyle [x] + [x] + [x] + [x] + [x] = [5x] =[0] $ so that $\displaystyle x = [0] $?