# congruences

• September 5th 2009, 10:20 AM
Sampras
congruences
Suppose you have equations like $x+x+x+x+x = 0$ in $\mathbb{Z}_5$.

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

Also the $x$ is just a symbol so could we just do the following: $[x] + [x] + [x] + [x] + [x] = [5x] =[0]$ so that $x = [0]$?
• September 5th 2009, 10:59 AM
ThePerfectHacker
Quote:

Originally Posted by Sampras
Suppose you have equations like $x+x+x+x+x = 0$ in $\mathbb{Z}_5$.

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

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

Right they all work.

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