# Thread: congruences

1. ## congruences

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 $\displaystyle x = [0]$?

2. Originally Posted by 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 $\displaystyle x = [0]$?
Right they all work.

Also, $\displaystyle [x] + ... + [x] = [5x] = [0]$ since $\displaystyle 5x$ is always divisible by $\displaystyle 5$. Therefore, any $\displaystyle x$ will solve $\displaystyle [x] + ... + [x] = [0]$.