# congruences

• Sep 5th 2009, 10:20 AM
Sampras
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] \$?
• Sep 5th 2009, 10:59 AM
ThePerfectHacker
Quote:

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]\$.